As you said, he used it even without eyes so what makes you think he specifically used it with his Rinnegan? His use of it isn't dependant on the eyes so there's no telling that he used his eyes for it when he had them.
And even if he did, that's one ability. With Genjutsu he needed to switch.
It isn't specifically tied to his Rinnegan, yet having Rinnegan doesn't make you entirely unable to Susano'o either. You originally stated that "Tomoe-Less Rinnegan can't use Sharingan abilities" which is inaccurate in this case as Susano'o is first unlocked through the Sharingan when it matures into a Mangekyou Sharingan by releasing special chakra from their brains. Tobirama's statement gives us some interesting information regarding the evolution of the eyes.
For one we know that the power lies in their brain, not their eyes. This explains why only a pure Uchiha is able to use the eyes effectively and why the original owner is the only one able to use their own eyes to their fullest potential. The closer they are to the original owner, the closer the "link" between the chakra in the brain and the chakra in the eyes. We also know that the eyes themselves possess some power, hence my argument that when an Uchiha reaches a high emotional stage they physically mutate.
Upon losing something that means a lot to you, or being overflown with emotion of something like love, the Uchiha releases a special chakra from their brain that causes this mutation into the Sharingan. At this point the Sharingan has become a "vessel" of sorts for that chakra, an extra storage area in their body. When the Sharingan is activated by the user in combat, the chakra in those "vessels" are activated, resulting in the change in color and pattern. Nevertheless their powers such as Susano'o don't originate from their eyes, but their brains.
However, when a user loses their eyes, they also loses a lot of their chakra that was stored to be used with their eyes. However they still hold the power to use their powers somewhat as seen with Madara. This allows them to use a weaker variant of their full abilities or in certain cases not use their abilities at all if they rely on eyesight, which all known techniques excluding Obito's "phase" ability does. Also, what Genjutsu are you referring to? Infinite Tsukuyomi? That can only be case with the Rinne-Sharingan, not the Rinnegan, but it can be countered by the Rinnegan and Bijuu chakra.
Even Obito without Rinnegan could control and Gedo and with it he was an absolute beast and could actually employ the Gedo in combat. And Nagato's use of it pales in comparison to Obito's display. The gedo turned Nagato into a cripple, while Obito had it as his plaything.
What? Where did you get that from? Obito never controlled the Gedo without the Rinnegan, in-fact he went and got the Rinnegan specifically so he could use the Gedo in the war. When Obito and Zetsu first approached Yahiko, Nagato and Konan the following was said by Zetsu
Nagato is of Senju Bloodline. And the only other person in addition to Madara who can summon the Gedo Statue.
- Black Zetsu
While Madara himself stated the following to Obito when he explained the plan and the requirements for it to work properly:
The Rinnegan cannot be awakened without possessing both Uchiha and Senju powers. And the Statue cannot be easily handled. You have Senju Cells attached to the right half of you... Even if you never awaken the Rinnegan, you ought to be able to control the statue.
- Madara Uchiha
The two combined put out a few things. First is that the Rinnegan is required in order to
summon the Gedo Statue. The second is that in order to fully control the statue, one needs the powers of both the
Senju and
Uchiha. As mentioned earlier, Obito wasn't in Madara's plans originally, and him having Senju DNA over half his body certainly wasn't part of the plan. It was done on a whim because it was "good enough" and Madara didn't want to wait anymore. He even admits that he should perhaps have waited until Sasuke was born.
So this means that in order for someone to control the Gedo Statue in order to gather all the Bijuu, someone who had Senju Power was required as Madara would give them the Uchiha powers through the Rinngan he would give them as they needed that in order to summon the Gedo Statue itself. Which is why Nagato was chosen as any other candidate was likely too old or too difficult to manipulate.
This is nonsense, he mentioned he could barely handle one's power
You know you just called my quote "nonsense" then went ahead and re-quoted the exact same thing word for word, right? Anyhow. It isn't nonsense as you demonstrated with the panel.
Obito has Senju DNA, rokudo rods and has Uchiha blood, yet this is what was happening to him. He even mentioned he
couldn't implant 2. This clearly tells you Madara's eyes power is too strong even for Obito with his buffs and blood.
As the panel (and my quote said earlier) says, he was
starting to lose himself, he hadn't done so yet but could feel he was slipping away and being overtaken by the power with just one eye. This implies a progress, not an instant loss of oneself. Meaning it is not clear whenever or not Obito would be able to control both eyes for a set amount of time or not. Judging by his wording its more likely that he would be able to use both, but for a severely limited amount of time before losing himself entirely to them. Neither scenario is made clear by him, only that he didn't want to risk it by taking both eyes as he could already start to feel the side-effects of one.
And Sasuke confirming anybody(non Uchiha or senju) could use his tells you his is inferior.
This... Isn't accurate and is overreaching. Sasuke doesn't explain if there are any side-effects to this like what Obito explained. As he doesn't explain it and we don't see it, then its unclear whenever or not it would have side-effects or not. Hence it isn't set in stone and at best would be a theory. That being said, we know that Madara's Rinnegan was a "deviation" of Hagoromo's chakra where as Sasuke's is the pure chakra of Hagoromo. Both Rinnegan also possess the same amounts of powers, each having their own unique power in form of Limbo and Heavenly Hand Power. The only difference in display between the two different variants is the size of each technique displayed.
For example, Madara created more and larger Chibaku Tensei meteors than what Sasuke was able to do. This argument doesn't hold a lot of water as it can be explained in more than one way. Such as Madara simply being in possession of more chakra and we know from Nagato's statement that performing the Chibaku Tensei, a massive amount of chakra is required. Another argument is that Sasuke simply didn't need bigger seals on the Bijuu than what he did. Hence he didn't make them bigger. Point being, you can argue away this one instance in multiple ways, meaning it isn't a good source for evidence against another argument.
It's true that Susanoo can be used without eyes, but that's a Madara feat. As for the perfect Susanoo nobody has used it without eyes, and Sasuke linked his inability to use it to his chakra.
I assume you are referring to Sasuke from the Seventh Hokage and the Scarlet Spring scene where he comments;
I'm taking all of us there. Since my ocular powers are finally coming back. Searching for and entering Kaguya's time-space... Requires a lot of chakra.
- Sasuke Uchiha
At which point Sasuke activates Susano'o. This isn't something new though. The use of techniques require chakra and if you lack said chakra you cannot use said technique. That being said, even without his Tomoe and EMS being able to activate against Shin, he was still able to use Heavenly Hand Power to teleport in the short fight with Shin. We also see during his fight with Naruto that even with his Rinnegan and chakra in a weakened state he was still able to use Preta Path. Hence these abilities either don't require a lot of chakra or he still had more than enough chakra for them to be used.
On top of that, the Susano'o is the user's chakra that has taken physical form. It is a Ninjutsu technique that is derived from the Mangekyou Sharingan specifically. When Sasuke mentioned he was out of chakra in the Scarlet Spring, he was also incapable of using his Eternal Mangekyou Sharingan, only using his Three-Tomoe Sharingan. Upon regaining chakra his tomoe in the Rinnegan reappeared and his EMS turn on again. This falls in line with what we know about the Sharingan, that it drains chakra from the user.
Likely to replace "its storage" as it is being used and the more chakra you have and more control you have over this chakra, the less of a problem it is unless you use extremely taxing technique. That being said, we know for a fact that the Sharingan "drains" chakra from the user. This implies that the Sharingan has some form of chakra itself and uses that up, thus needing to be refilled like a glass of water. This is filled from the user's own pool, which can result in death if over-used just like any Chakra techniques. This is also backed up by Tobirama's statement that the chakra to manifest the Sharingan originally comes from the brain of the user.
This in turn means that the chakra that is being used to do techniques such as Amaterasu or Susano'o comes from the brain, not eyes. We can theorize as to why a given eye can use a given ability though. We know that the Uchiha doesn't have to preform handsigns to use their techniques, only doing so when they want to control them better or use them at a higher level. This means that at some point inbetween their brain and eye, their physical and spiritual powers are molded together naturally and thus don't require handsigns to focus that effort.
Hence the eyes likely work as an automated function that molds the chakra into a specific technique as it drains it from the host. Which would also explains the user's inherit knowledge of their techniques from the moment they get them. As Susano'o isn't tied to either eye, but both, and it is just pure chakra then its more than reasonable to say that anyone who has the chakra required to use it can do so without their eyes. Though without having their eyes as a "focus" they power becomes far weaker.