Good thing there is nothing in the game that forces a player to only have +/-1 for their "bad" save(s), and nothing in the game that forces a DM to use creatures with DCs of 19+ on their abilities.
There are no issues because we can choose to ignore them? Arguably more true for 5e than for any version since the early days. I could absolutely choose never to use same-level monsters that forced saves as written, it's not even much of a burden, as I frequently change up monsters, anyway. A fighter could invest all his ASIs in CHA and WIS, and get his saves all the way up to +4 in each (or, if feats are on the table, use them to gain multiple save proficiencies), while leaving his STR at 16, and his Indomitable would just manage to mean something when some 20+ CR monster tried to dominate him.
Still not so true that I wouldn't consider a simple variant to 'fix' the problem over just pretending it wasn't there.
The truth of the game, Aaron, is that loads of player characters will have a -1 or 0 in their weakest save, and that the rules stipulate that any level 20 spellcaster with ability 20 will have a DC of 19+. And so the game *will* feature saving throws you can't make.
Well, if we assume array...
15, 14, 13, 12, 10, 8
and no feats...
All classes get at least 5 ASIs, right? 5 ASIs is a +10 distributed over 6 stats. Assume you had a +2 racial in your highest stat and want to max it. That's 15+2= 17 + 3 f/asis is maxxed at 20. You have 7 points left. 1 to round up that 13.
20, 14, 14, 12, 10, 8 and 6 points left.
2 to the 10, 4 to the 8.
20, 14, 14, 12, 12, 12
Not 0 or -1 save. One +5 (probably +11 at 17th, because it's likely your proficient in your primary, one at +8, and the rest go: +2, +1, +1, +1). That's a pretty obsessively CYA stat distribution. There are a few DC 22 and 23 saves in the MM. There's also some +1 save items in the DMG.
Doesn't sound like untouchable save DCs are quite inevitable. They can happen. Of course, the DM could always set a DC as high as he wants, or narrate failure on a save without calling for a roll, or create an effect that gives no save.
Whether it's a significant flaw, and if not fixing it had other, positive, results, are different matters. But it's a flaw.
It also seems to me that it's a fairly easy fix. Even the 'just don't use/just reduce higher save DCs' suggestion addresses it, if not ideally.
I'm increasingly thinking that it'd be a good idea, not just for saves, but across the board, to change the +2 to +6 over 20 levels proficiency progression of 5e Bounded Accuracy to +0 to +4 over 20 levels across the board (all rolls, maybe even AC), with a static +2 for proficiency. Stated exactly that way it does mute Expertise, but I'm not sure that's a terrible thing, and it only directly impacts two classes...