I think that chemistry 101 classification of bonds is a tad useless here. Instead, you can go from first principles: there are things that happen when atomic orbitals overlap (covalent bonds, metallic and such), there are interactions that are mostly electrostatic in nature (ionic, dipole-dipole, quadrupole-quadrupole - important biologically as pi-stacking, also ion-quadrupole etc) and there are things that are a result of exchange interaction (van der Waals and steric repulsion). Hydrogen bonds would be a mix of dipole-dipole and van der Waals interaction. You don’t have to transfer electrons in order to have ionic interaction, most of the time in biologically relevant situations it’s proton transfer, or charges just were there previously. Hydrophobic interactions are almost entirely a solvent effect and aren’t a bond strictly speaking

In water, i’m pretty sure that proteins are mostly held by hydrogen bonds and hydrophobic interactions. EY is correct in that some proteins hold shape by mostly noncovalent interactions, but these are mostly hydrogen, ionic, hydrophobic interactions and the proteins that actually provide mechanical strength run in continuous covalent strands through entire length of them anyway (collagen, keratin). I don’t think that counting bonds and saying that something is 90% bound covalently is a meaningful metric, because long series of hydrogen bonds or even vdW forces (in things like UHMWPE fibers) can be stronger than single covalently bound strand, ie if you tried to pull out a single strand of kevlar or collagen from bulk material, above certain length you won’t pull it apart, you’d just break it because collective energy of hydrogen bonds will be greater than single covalent bond holding it together, that’s why these fibers are strong in the first place

There is another kind of flexibility that you haven’t mentioned: proteins are made out of single covalently bound strand, yes, but these aren’t straight C-C chains. Making and especially breaking C-C bonds in controlled way is hard, proteins can be just hydrolyzed at amide bonds. If protein breaks in some way, and in real world everything breaks, it can be recycled into aminoacids (+ any cofactors etc) and then put back in a pretty straightforward way; you can’t do this with diamondoids, when it breaks, it breaks hard, and you’re done unless you’re picking everything apart atom by atom which would be much harder and more energy intensive. As it happens you can buy bulk adamantane, but it’s just made in conditions where C-C bonds are weak (high temperature) and it’s preferentially formed because it’s most stable thermodynamically among its isomers (that are starting materials). Conversely, if you use weaker bonds, you can make pieces conform to some template, or to each other without breaking everything at once - this is basis of dynamic combinatorial chemistry. There’s also entire field of self-healing materials that is based almost entirely on these either noncovalent or reversible covalent bonds

As part of the process, parts of the enzyme actually shift and mold their structures around the incoming molecules in order to better catalyse reactions. I’m not sure how easily you could replicate this using stiff strictly covalent structures.

You actually don’t have to do that, and there are some small organocatalysts that are entirely covalently bonded and do the same job. However you can’t make them from from aminoacids, these don’t have secondary structure (too small) and are generally less active. The bare minimum is to provide a receptor for transition state, and you can make it work without drastic changes in conformation. You could make your catalyst as stiff as you like, and it’ll even make activity higher - but only if none of these stiff parts interfere with binding of substrates, and your options are limited. It’s often better to leave some wiggle room. Short peptides aren’t really stiff enough in ways that matter there and instead it’s secondary and tertiary structure that puts important bits in the right place

Feel free to pick my brain any time you like