Hamura and hagomoro fight

[Raidori]

In the flashback of their fight against each other because of kaguya, hagoromo used the seal of senjutsu on hamura who had the byagukan. Borutos dojutsu had to occur because he has a mix of senjutsu, kurama chakra(hagoromo) and byakugan blood. Hamura s eye must've awakened slowly after he left for the moon and because of the senjutsu he was able to acquire the moon staff as well as an early power up, as his powers matured he awakened the tenseigan

 

[Glad Of War]

The fact Toneri awakened the Tenseigan without having Senjutsu disproves your theory.

 

[The Great Second Hokage]

The fact Toneri awakened the Tenseigan without having Senjutsu disproves your theory.

Wrecked

 

[Raidori]

He awakened tenseigan on the moon after the fight with kaguya which was after the seal was placed on him that gave him senjutsu

 

[InfiniteMugen]

Wtf did I just read

 

[Rikudou Tobi]

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[Yahcob13]

I think byakugan/tenseigan users have a way of absorbing energy in a way similar to senjutsu. The moon shinobi were using it as a power source.