All sorts of things you care about could depend on a single result. Your life, an allies life, whether you drop the McGuffin in the volcano it was forged or meekly hand it to the BBEG who just dominated you.
None of those things depend on a single roll.
Evidence? Just an illustrative example of how a bad save can snowball. And, not necessity. The question wasn't "Is it absolutely necessary to save 5e from itself by giving everyone save proficiency?"
Okay, but it wasn't a very good example because
in practice, it doesn't snowball, thanks to making a saving throw not being the exclusive determiner of success or failure.
It really, really does. Especially the 25 years prior to the last 15, if that makes any sense. ;P
Not in spell mechanics. 5e's
dominate spells, for instance, are TREMENDOUSLY more easy to counteract than 2e's or 1e's or OD&D's (while remaining evocative of what those spells could do):
2e Domination: As a 5th-level spell, make one save at -2 (adjusted for Wisdom) or be dominated forever. Get a new save with a bonus if "forced to take an action against your nature." A "typical 2e fighter" of ~10th level (when the casters get this ability) needs a 13 or better to make that initial save. Basically: if you fail this save, you are an NPC's puppet forever.
5e Domination: As a 5th-level concentration spell against humanoids, make a Wis save or be charmed (with advantage if combat is going on). Caster can give you general directions that you can interpret, or can spend their action to take precise control. If you take damage, you make a new save. The "worst case scenario" circa 10th level is a Wis save at -1 vs. a DC 16 (17+ succeeds). If this was going up against a champion fighter, they'd have indomitable.
Our low-WIS Fighter is probably going to get hit in 5e (even with Indomitable), barring some significant luck. But the effects once hit are
remarkably different, and 5e-domination is much, much easier to get out of, and even limits the caster while they're doing it.
Yeah, since 3.5, D&D had been increasingly going the direction of not utterly ending a character on a single failed save, and 5e didn't back off that as much as it did some other recent trends. That doesn't mean a low level character having a +1 instead of a -1 to his worst save is going to break the game, or even register in the very loose balance of 5e. Nor will a +5 vs a -1 devastate playability at high level - even assuming high level gets played any more often in 5e than it did in classic D&D.
I mean, a +6 bonus is
significant. Folks will make most of their saves, even the ones they "suck at". The only question is if you're cool with that. It doesn't appeal to me much, since I like a game where things like not giving the McGuffin to Greg the Wis 8 Wonder-Warrior is a choice with significant meaning and finding out that the BBEG can dominate people is important intel and the vagaries of chance might hose you on an adventure. I like that a LOT MORE than a game where the PC's basically steamroll through stuff because they're free to ignore their dumpstat.
In classic D&D, a very high level character would pass saves very easily, often on anything but a natural 1, so saves being vs horrific things still didn't intimidate them that much. In 5e, failing a save is not always so gruesome a fate - but failing because you rolled 11 is still very different from saving because you rolled 1 - it doesn't need to be failing because your rolled a 17 to make up for toning down the consequences.
Failing because you rolled an 11 or even a 16 -- when
you deliberately chose to play a character with this weakness -- is something I'd expect. You should fail a lot when someone hits you where you're weak. That's part of what makes interesting choices and consequences. You rely on the rest of the party to help you out in the areas where you're not as strong. That's also what I'd expect from a game that features teamwork.