Are these proofs equivalent? Pretty cool if so.

No, they do not seem to he equivalent. Not a mathmatician but running the Deepseek proof through ChatGPT gives:

"If everything is made rigorous:

You would have a valid independent proof It would contain real structural insight It would not replace the flow proof as the “best” proof

But:

It would still be a meaningful alternative proof with explanatory power, not just a redundant one."