as soon as dsm was revealed my thoughts matched these, someone had to teach the old frog/snake/slug the SM, and So6P was known for being a Sage but no specific animal+Rinnegan animal path multiple summons lead me to believer So6P had the 3 (or more) old great sage's as his summons and was the one who taught them(the age of the frog is testament to this)
I think So6P is viewed by readers for his Rinnegan/Mokoton and being the 10-tails Jinchuuriki, people tend to forget that he was a Sage before 10-tails, and beat 10-tails probably with Sage abilities..its usually over looked because there is nothing specific about the reason hes a Sage, and is assumed he was called a Sage because of his massive power
He is the one who created Ninjutsu, so its probable that he possessed all of the Senjutsu there was and probably "split" it between his summons, so to prevent someone gaining that great of a power and using it for destruction
Seems like everything he left was split;
Eyes and Body
10-tails split into 9
Sage mode split into 3(or more?) via teaching his great summons a piece each
I mean you gotta think, he had the greatest terror sealed within him, an evil much heavier then 9-tails and as far as we know he didn't ever struggle with it taking over his body etc, so this indicates he had absolute harmony between Good/Evil so it makes sense that he could use all of the Sage Modes in harmony, being able to keep the balance of good and bad when such a massive evil is sealed inside of him is testament to his ability to use the different Sage Modes in unison
You can argue that each Sage Mode is extremely hard to learn and possessing them all is overkill, but remember it was said by either Kakashi or Jiraiya that Rinnegan users could master Jutsu in very short amount of times, plus So6P created Ninjutsu, and Senjutsu is a form of Ninjutsu in that aspect, so its very believable that So6P had all of the Senjutsu
It didn't take Naruto too long to learn Toad SM, imagine if Nagato had been given that chance, it would of been shorter for him to learn it because of Rinnegan
tldr; yes i agree there is a "true sage" mode that contains all of them in harmony, and so6p probably used this more often then we think, as he was known as being a Sage more then anything, even more so then being the 10-tails jinchuuriki, backed by the shock of Kages when Tobi told them the history of So6P