First Contact bootstrap paradox

Hello! Trek fan my whole life, follow a handful of YouTubers, not so much forums. I was re watching first contact today and it occurred to me, is there any (even if non-canon) comments that point to the possibility that: At the start of the film, Picard sees the damage and says “we have the schematics on the ship” and we later see the crew working on it. What made me consider the next part is when Barclay comes up with a copper tube, and Geordi scans it with his eyes, and suggests some technobabble upgrade.

And therefore: the original Phoenix may have never worked, and that only the bootstrap paradox schematics from the Enterprise allowed it to function properly?

Edit: Also, with the AU in the final season of lower decks, a failed warp test without the Enterprise’s plans could’ve lead to the alternate form of travel they invented. (Trying to avoid spoils I guess)