I dunno... have you had trouble with this in actual high-level play?
Not even theoretical trouble in 5e, since I would just house-rule the save progression anyway. The 'trouble' in question, though, was painfully evident in actual experience with 3e, and stood in contrast to my experiences with 1e/2e in which saves improved as you leveled, across-the-board.
Sure, there are six saves, but they aren't all equally important. I'd expect a really high level character to have pretty good scores in the most important saves.
There are three 'important' saves (though really, STR saves aren't that rare, and failing a rare INT or CHA save can have serious consequences), most classes are proficient in only one of them. If it doesn't happen to align with their prime requisite, that's 4 stats they have to boost as they level, so either they neglect one of those entirely, or most of them go up by only a point or two over 20 levels. That's unlikely to get up to 'pretty good.' Now, if feats are available, you can shore up several saves, at the cost of lower stats overall...
And those low-level monsters have pretty low DCs (mummy is DC 11), and they probably wouldn't even get a chance to use their abilities. That mummy would get destroyed before it got within 60 feet of any of the PCs, probably.
Also... in my understanding, low-level monsters are supposed to remain relevant longer in 5E.
Bounded Accuracy - that +4 from proficiency over 20 levels - leaves them relevant. An 8 WIS character would be scared by that mummy on an 11 or less natural roll, 55% of the time. Under the variant I lean towards (prof-2 to 'bad' saves vs full prof for good ones), at 20th, with the same 8 WIS, he'd save on an 8, failing 35% of the time - the monster would still be relevant, but at least the character would have experienced some advancement.
This seems to be a really good summary of the objection to how saving throws work in 5e. While I don't share the objection, it does make me have to think through why it doesn't bother me, because what you're saying there is the kind of thing I would normally agree with.
While I haven't been able to totally figure it out, I think the main reason it doesn't bother me is that almost all characters gain some way of resisting or mitigating the effects of magic through their class advancement, even when it isn't a simple equivalent of more hp (numerical).
I hope you're not thinking of things like 'Indomitable...'
Classes with features that can help with any/all saves:
Bards, Clerics, Druids, Sorcerers, Wizards: Spells (helping directly or indirectly)
Fighters: Indomitable
Monks: Proficiency in all saves
Paladins: Charisma to all saves
Proficiency in all saves works, I'll grant. I'd be satisfied with a +4 over 20 level advancement on all saves, and that's probably what I'll do when I run a campaign to high-level (if that ever seems like a good idea - saves aren't the only thing that make me hesitate to do so, for instance, I feel that running introductory games in organized play does more to help the hobby in the long run). Indomitable is a bad joke without some improvement to saves from leveling, run up against a DC 19 in a non-proficient save, it gives you the privilege of failing twice. But I'll grant that casters can mitigate the problem - and benefit from it more than they suffer, because they can target enemies' bad saves.
Classes with one notable weakness (of the big 3 saves):
Nod. I get that a save weakness differentiates classes. I just don't think it has to leave 20th level characters as bad off as they were at 1st level to achieve that differentiation. Different primary/secondary stats and a lesser difference in proficiency bonus would be quite adequate.
Have you considered that perhaps it is wrong to believe this to be true in 5th edition?
Dump stats are still inevitably a thing in 5e. Ironically enough, especially if feats are used. I give 5e every possible credit for trying to reduce that with countervailing incentives - the current 6-save topic, the wide range of ability checks that can potentially be called for. But, yeah, dump stats are still a thing, and MAD is still a disadvantage. Short of just up and giving all starting PCs straight 14s, I don't think it's going to change.
Is that what is being asked, or might the game be asking the players to fail some saves and find another way to mitigate the effects of failing a save?
Clearly when an untouchable DC is set, the intent is for the PC to fail. In the rough 'spotlight' balance of 5e, that's not entirely unexpected, just a bit heavy-handed. And it's something the DM can deal with on a case-by-case basis. Just call for a DC 17 instead of the 23 in the
module adventure. I'd just give a scaling bonus to non-proficient saves and take care of the issue that way. 5e may have 'problems' like these but it invites you to solve them.