>other techniques include Kant's Transcendental Deduction or Descartes's pure doubt

This is not quite accurate. Kant says very explicitly in the (rarely studied) Transcendental Doctrine of Method (Ch 1 Section 4, A789/B817) that this kind of proof method (he calls it "apagogic") is unsuitable to transcendental proofs.

You might be thinking of the much more well studied Antinomies of Pure Reason, in which he uses this kind of proof negatively (which is to say, the circumscribe the limits of reason) as part of his proof against the way the metaphysical arguments from philosophers of his time (which he called "dogmatic" use of reason) about the nature of the cosmos were posed.

The method he used in his Deduction is a transcendental argument, which is typically expressed using two things, X and Y. X is problematic (can be true but not necessarily so), and Y is dependent on X. So then if Y is true, then X must necessarily be true as well.

Sorry I meant "proof method" as more like "this was this guy's angle of attack", not that they would've thought each others angles were valid at all or that they're commensurable with say, 20th century formal proof logic (or Aristotelian logic for example). Descartes and Leibniz were squarely the rationalists that Kant wanted to abolish, and Hegel rejected Kants distinction between noumena and phenomena entirely, so they're already starting from very different places.

I guess it would be more accurate to state Kants actual premises here as making the distinction between appearance and thing-in-itself rather than the deduction, but the deduction technique itself was fascinating when I first learned it so that's what I associate most with Kant lol.

I guess I have not thought critically why we couldn't use a Transcendental argument to support Descartes. I just treated it as a vague category error (to be fair I don't actually know Descartes philosophy that well, even less than I know Kants lol). Could be a fun exercise when I have time.