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As someone who used to work in peptide synthesis, just forming an amino acid sequence is the first hurdle (and a fairly low one). Some synthetic couplings simply refuse to proceed, presumably due to the growing peptide chain folding in a way that makes the terminus chemically inaccessible (which is dependent on the side chains carrying protecting groups, so not a function of the final sequence, but still sometimes consequential). You also have issues of solubility in the final sequence. If your peptide is insoluble in water then its practical applications may be very limited (though you can often get them to dissolve in organic solvents).

Beneath all this hype about engineering proteins is the assumption that nature hasn't already thoroughly surveyed the protein sequence landscape. It is easy to do calculations which give "bigger than the known universe" answers (like 20^100) but nature doesn't explore protein structure on those scales, and it doesnt need to produce every protein sequence simultaneously. Instead it scans for functions on shorter, less complex sequences where the side chain/backbone angle tendencies coalesce and then builds together those short sequences into higher structures. I suspect very little potential protein functionality has been missed in the last few billion years of trial and error that humans can simulate our way into discovering.

There are also only so many types of chemical reactions and proteins catalyse all of them. Adding different amino acid side chains could circumvent the need for catalytic cofactors, but that feels like a step backwards to me, adding unnecessary complexity to a system which already works well.

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"There might be a minuscule subset that forms bizarre reactive loops or self-hydrolyzing sites (of which I can find little information on what consistently causes such sites"

somewhat tangential but I'm now pondering whether inherently explosive proteins might be possible via post-folding reactions, similar to how GFP forms a naturally fluorescent chromophore via post-folding cyclization

don't know nearly enough chemistry to guess whether any plausible routes to this exist using natural side chains tho

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or, if not via side chains, maybe some sort of crazy self-modifying enzymatic reaction on glycans that results in something like nitrocellulose...

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