It makes perfect sense. Obtaining Hagoromo's chakra doesn't give you Ashura and Indra chakra. It gives you Hagoromo's chakra. Ashura's chakra and Indra's chakra can be combined to recreate Hagoromo's chakra.
Ashura's Chakra = Hagoromo's Physical Energy
Indra's Chakra = Hagoromo's Spiritual Energy
Hagoromo's Physical and Spiritual Energies are equivalent to Indra's and Ashura's chakra and these are essential constituents for composing chakra. This is a fundamental aspect taught to us at an elementary school level ever since Part 1. Trying to argue that Hagoromo's chakra didn't have these constituents despite the fact that they're equivalent to his Physical and Spiritual Energy is pretty ridiculous.
If you obtain Juubi chakra, you don't have Kurama chakra. You have Juubi chakra. Even though Kurama's chakra is combined with other chakra to make the Juubi chakra, you still don't have it. The manga clearly distinguishes between different chakras and makes it clear that Juubi chakra is not the same as Bijuu chakra. Same goes for individual people.
No, they literally have Kurama's chakra which was explicitly shown when Hinata sensed the the constituents of the Juubi's chakra enter Obito's body or when Obito snatched Kurama from Madara's grasp. They still possess Kurama's chakra and quite frankly, I don't see how this pertains to the discussion.
Hashirama's cells from Kabuto didn't contain Ashura's chakra, because Ashura transmigrated from Hashirama to Naruto, after Hashirama died. Therefore, his body and cells no longer contained the chakra of Ashura.
Bruh ... Even though Hagoromo stated that
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Naruto does not have Indra's chakra either. :lol
Databook 4 said:
Here we will attempt to analyze the primal properties of the mysterious Yin and Yang Release jutsu.
Six Paths
The secret jutsu that created all things from naught.
Yin Release which can create form out of naught and Yang Release which can instill life into that form. Deftly manipulating these two properties, the Sage of Six Paths—Ohtsutsuki Hagoromo—developed the Creation of All Things Jutsu. He likewise created the nine bodies of the Bijuu from Ten Tails' chakra, making them another gift of Yin and Yang Release. Later, the Uchiha Clan would tend to inherit the power of Yin Release, while the Senju Clan would tend to inherit the power of Yang Release.
↓ Mental and physical energy. That is Yin and Yang Release.
↑ Having gained Hagoromo's power, Naruto was able to perfectly demonstrate the power of Yin and Yang Release.
1. Hagoromo possesses both Yin and Yang Release.
2. Uchiha possesses Yin Release via Indra who received Yin Release via Hagoromo.
3. Naruto possesses both Yin and Yang Release.
Honestly not even that hard to understand that he has Indra's chakra since it's explicitly stated that he has both Yin and Yang Release. Not sure where you guys arrived to the conclusion that Naruto only has Hagoromo's Yang when Kishimoto's words via Databook suggests otherwise.