Oh, no doubt about it, Shinji himself is capable of triggering Evas into an awakened state. He's called an Impact trigger for a reason.
But to go back to my earlier point: Shinji by himself isn't capable of restoring an Adam!Eva to its Radiant Giant state. Absorbing an Angel is required for that. I'm unclear what part Lilith's "factor" plays in things, so, setting that aside, the following explanation is possible:
We saw that, at Second Impact, the Adams were all in Radiant Giant state. Something happened involving the Spears to stop or suspend Second Impact. Three of the Adams seem to have physically survived, albeit in a "burnt out" state; but they are capable of "pseudoevolving" back into Radiant Giant form via a combination of (A) having the right soul linked to their core via entry plug, and (B) absorbing an Angel, a being with the Fruit of Life. (B) seems to indicate that the Adams lost their Fruits of Life somehow, and are able to get them back from Angels. What about the soul part, though? It's been suspected for some time that the Eva pilots (minus Rei) actually have the Adams' souls. With Kaworu, this is practically a certainty, because, well, Kaworu; but Shinji's weird "Impact trigger" status can be readily explained if he has an Adam's soul. It's quite possible that the same holds for Asuka and Mari, and that they must also wear chokers because of it. (A production illustration by Yamashita shows Asuka, at least, clearly wearing a collar of some sort.)
The final piece of the puzzle here is the Adams' Vessels. Eva Mark.09 is, according to its full designation, only one of them -- iow, there are others that have existed or still do exist. "Vessel" is utsuwa, which means "container". The only other time (that I know of) that this term is used in NTE is in the Explanation of Eva 1.01, where the entry plug is termed the "Vessel for a Soul" (that of the pilot's). Mark.09's true form is that of a polymorphic core, and cores are things that contain souls. You can maybe see where this is going. The Adams' Vessels are probably the disembodied cores of the Adams, separated from their bodies in order to bring Second Impact to an end. This would have separated the Adams' bodies from both their souls and their Fruits of Life, explaining why they need to regain both in order to be "repowered". This also means that the Adams' Vessels and the Vessels for Souls (= Entry Plugs) are different containers for the same things.