How did EVA Unit 01 interfere with SEELE's plan?
So, after reading the manga extensively and going into detail about the meta of the episodes close to the end of anime, I think I finally pieced together the reason why EVA Unit 01 was grounded by SEELE, and how EVA Unit 01 interfered with SEELE's plan.

In the manga, EVA Unit 01 was grounded by SEELE as a form of "insurance" for executing their version of Third Impact, only to be used in case Gendo Ikari executed his version of Third Impact. EVA Unit 01 was grounded based on the fear that Lillith might become compromised should Gendo go ahead with his version of Third Impact.

This also explains why EVA Unit 01 is the only Evangelion Unit to made from Lillith as opposed to being made from cloning Adam.

I think that EVA Unit 01's awakening phase was not accounted for correctly by SEELE in their master plan, and therefore was grounded to readjust and incorporate an awakened Unit 01 into executing Third Impact. I also think that SEELE was afraid that both EVA Unit 01 and Lillith may be compromised, and thus tried to control what they could from the shadows as best they could, whilst still ensuring Third Impact (i.e. grounding EVA Unit 01)