Even without Title 42, the President can control the border: I wrote about Section 212(f) a few years ago in a slightly different context. See https://www.americanthinker.com/blog/2017/02/the_law_that_never_was.html
See this discussion:
https://cis.org/Arthur/If-You-Title-42-Youll-Love-Section-212f
JKP's words "quickly removed" does not seem to be the exact language in the federal code, but even if it was, it would be useless. "Quickly" might mean 10 years, but even it there was a limit. there would be no penalty or remedy for not meeting that limit.
In fact, the Biden gang is determined to import more invaders than "our system" can possibly support. Don't hold your breath waiting for SCOTUS to do anything useful (like requiring federal agencies to stop allowing entry based on a non-citizen's declaration).