I think there is an important way of looking at questions like these where, at the final end, you ask yourself, “Okay, but does my argument prove that flesh is in fact as strong as diamond? Why isn’t flesh as strong as diamond, then, if I’ve refuted the original argument for why it isn’t?” and this is the question that leads you to realize that some local strong covalent bonds don’t matter to the argument if those bonds aren’t the parts that break under load.
It’s not clear to me that covalent bonds aren’t the ones that are breaking under load when talking about flesh though.
Covalent crosslinks (such as the disulfide bond you mentioned earlier) aren’t merely an irrelevant edge case, proteins like collagen (which is used in the extracellular matrix and connective tissue) and keratin (used in hair, nails, horns and hooves) also have such crosslinks.
A short and oversimplified answer is that the keratin in horn is not as densely linked with bonds as diamond is, and consequently the atoms are less confined (in a way diamond is sort of like a maximally crosslinked material, though it’s not usually described that way).
Generally speaking, crosslinking polymers (including proteins) tends to increase their rigidity. To use a non-living example, when latex is treated with sulfur, the polymer chains also get crosslinked with those same disulfide bonds, producing “vulcanized” rubber which is harder and tougher.
The crosslinks are why you’ll sometimes see people say that vulcanized rubber is “one big molecule” (though in practice it’s hard to tell if the crosslinking was actually so complete and to link every polymer chain). This is also why vulcanized rubber doesn’t really melt, increasing the temperature will cause chemical changes instead (and while I’m not sure, my educated guess it that something similar would happen if you try to melt animal horns).
P.S. I didn’t bring it up earlier, but I don’t think your earlier statement about the way the masses of the atoms affect the bond strength is accurate. As a counterexample I’d point out that the deuterium-oxygen bond in heavy water is actually a little stronger than that of the protium-oxygen bond in regular water, and in that case the only change is using a more massive form of hydrogen.