For me, the virtue of the classic saving throws is that they divorce a big chunk of character competence and durability from how good your ability scores are. I think it's a mistake and a weakness of later editions to make saves dependent on ability scores.
Prior to AD&D, a character could be highly competent and badass with average ability scores. Once Gary made the charts dependent on high scores for bonuses (to the extent of saying that a character "needed" at least two 15s to be viable), it fed this whole escalation where ability scores became hugely important and a character with low or average ones was cruddy.
Making saves also dependent on them has the virtue of mechanical simplicity, but it further emphasizes their importance, which I don't think is a great move, unless maybe you want to have them increase reliably as characters advance in level like Tunnels & Trolls or WotC-era D&D. But for my tastes that still makes them too important.
Now, that being said, I think there's definitely room to tweak them. There is certainly something to be said in favor of more intuitive save categories than the classic 5. Honestly I think 5th edition took a misstep in going to saves for all six ability scores. For the physical ones it's usually easy to distinguish, but dexterity clearly rules the roost in applicability. For the mental ones, it can be hard to really intuit which one should apply for a given magical effect. Who's really to say whether breaking or resisting a mental enchantment should be a function of sharp wits, common sense, or force of personality?
The result in 5e seems to be that dexterity and wisdom dominate in importance, with Constitution getting a look in, and the others largely ignored.
I think 3E and 4E actually may have had the best approach here, with the broad categories of Will, Fortitude, and Reflex being easy to understand for nearly anyone.
4E mitigated the issue of ability disparity a bit by letting you use either Strength or Con for Fortitude, either Dex or Int for reflex, and either Wisdom or Charisma for Will. But I think I still prefer making saves be primarily level dependent, with minor bonuses for ability scores, racial abilities, magic items, and so forth adding in a bit where applicable.
Single save with modifiers definitely has potential too. You could even keep a little class differentiation, say by giving fighters a flat +2 against physical threats and hardships, magic users +2 against spells, and clerics +2 against supernatural threats. Thieves maybe +2 against traps and anytime quick reflexes would matter. You could do it many different ways, but that's just an example off the top of my head.