it's untenable because it shatters the entire idea of the setting being contemporary. contemporary combat necessitates the existence of contemporary firearms. it's really that simple.
in other words, you're leaving the scope of the premise. it'd be like proposing a fantasy setting without melee...
...or we can go the video game route and just ignore it.
why can blazkowics take dozens of 9mm and 8mm kurz rounds without dying? because he's a bad motherf-er, that's why. why does ulysses take so much punishment before dying? because he's a tough bastard, that's why. you don't really have to...
oh yeah, i forgot one:
for some reason when i think of monk for this i think of a gun fu john wick type. someone who marries martial arts with cqb gunfighting. then the barbarian can be your pit fighter or heavy weapons guy (or if you think a john wick type and room clearer type would be too...
the scripture-spewing preacher hunter comes to mind. yes that's more of a western trope, but it can work fine enough in the modern day.
or you can go full buffy the vampire slayer, i guess. "That Was Then...This Is Now" as a feature that lets you ignore damage resistances and immunities (EDIT...
i don't really care for "humans as quick versatile learners" tbh. i prefer "humans are cockroaches", but "humans are lynchpins of mundane reality" (draw steel touches that idea a bit) can also be fun.
that kind of flies in the face of 5e design philosophy (which is the only reason i didn't mention this - it's subjective whether or not it's really a "design issue"), but i don't really like that part of 5e anyway. even a level 10 character is a superhuman but the difference between your average...
saving throws. jesus christ these are a mess in 5e. too few proficiencies are given out for most classes and even the ones that scale don't scale enough towards higher levels. combine that with enemy saves getting ridiculous while your DCs scale super slowly and some saves being practically...
this made me think of moon knight's moon blades...which made me think of his moon plane, because i saw a video narrating a moon knight comic earlier where he described moon knight calling his ship to hit a running criminal as moon knight "throwing a blade at him", which i found really funny at...