Side benefit to saving throws is that they encourage off-turn attention.
And the game otherwise lacks a lot of off-turn action. Of course, getting hit and taking damage also happens off turn, much as one might want to ignore it...
This seems a solution in search for a problem. What issue, precisely, is being resolved by removing saving throws?
Consolidating attacks into a single mechanic is a clear system simplification. It has ripple effects, too. Not only is there one resolution procedure to learn instead of two, everything that interacts with offense becomes simpler. An effect that makes you better at dodging doesn't need to give attackers disadvantage to hit your AC and give you advantage to DEX saves, it can simply give attackers disadvantage to hit your AC or 'REF defense' or 'Touch AC' or whatever you call it.
I can't tell you how many times players have made an attack roll with Sacred Flame, or had to be prompted to make an attack roll with a spell that required one.
"Why don't we use THAC0?"
Because it was complicated and unintuitive, and would serve no purpose in a game where everyone would, in essence, have the same THAC0, and would shatter bounded accuracy like a crystal vase dropped from the back of an elephant - and stepped on by said elephant?
Why change that to be solely about the aggressor, and why would that be superior to just leaving it the same
Consistency & simplicity.
bearing in mind the effort required to rewrite whole chunks of the PHB as a consequence?
That's the rub, though, isn't it? In theory, it's simpler. In practice, it's too late to realize that gain.
Aside from it being a sacred cow, is there any actual need for saving throws? Is there any function that they serve that can't be served by a simpler attack vs. defence roll?
No & No. They're an artifact of the early game. Mathematically, you could invert all saves to be attack rolls (or vice versa), with no change in how often the target is affected. It'd be a significant simplification of the game, both in terms of new players learning it/simplifying play and in the sense of simplifying the writing of rules for anything that interacts with attacks/saves,
like Adv/Dis.
A related, interesting, variant, BTW, is 'player always rolls.' When attacking in any sense, a PCs make the attack roll, when being attacked, they make a defense roll (like a save). The DM never has to roll for his monsters. Of course, you could also invert that, and have the DM always roll, behind the screen, thus heightening tension and keeping DCs and monster capabilities uncertain.