Right. So one mechanic (Concentration) seems to conflate conceptual interpretation (or perhaps a better way is simulationist) while the other is purely game mechanic while ignoring more a more simulationist approach. In these two cases, I don't see how aligning both mechanics to either one or the other would be bad.
A difference is that death saves are something you
never want to be making, because the results of failure are
death, so the rules may be erring on the side of fairness or balance rather than integrity to the process. (There are definitely other games which use the equivalent of a Con check in order to stay alive in such situations). Concentration saves aren't
that big of a deal, and you might actually
choose to put yourself into a position where you would make one. By taking concentration to Con, it also gives spellcasters
something to care about, aside from just their one primary stat and maybe Dex. (Not that they didn't already care about Con, but it makes their second or third priority
more of a priority.)
This conflates one's physical ability with mental ability though. Pain is inherently a mental experience. Yes, it is grounded in one's physicality through nerve impulses and such, but the experience of pain is not necessarily physical. Also, look at monks who are able to balance for extended periods on a single finger or who can increase their body temperature through meditation. Implying that one's size can mitigate the impact of outside distractions on mental function doesn't make much sense to me.
If you throw a sportsball at a professional jockey, it will make a relatively larger impact than if you throw the same sportsball at a professional sumo wrestler. It's a greater displacement, which is why one is harder to ignore. You might not flinch if someone shoots you with a nerf dart, but you would probably flinch if someone shoots you with a bowling ball. Mass is relative, and Constitution is your Mass stat. A big fat guy with Con 20 has the same chance to keep concentration after taking 30 damage as a normal person with Con 10 has to keep concentration after taking 20 damage or less. (And if the idea of concentrating through 30 damage sounds silly, recall that the mechanic originally only allowed you to ignore like 1 or 2 damage, because they
also thought that ignoring the impact of 30 damage sounded silly.)
Monks are trained to ignore pain. To put it in 3E terms, they have Concentration as a class skill, even though it does nothing for them mechanically. In 5E terms, I guess you would say that they should have proficiency in Con saves - which they actually are, at high levels.