So, what are the design points of 5e saves?
1. PCs are strong is about 1/3 of their saves. Foes with varied attack types will sometimes match and sometimes bypass their strengths.
2. Foes are usually good on 1-4 saves, sort of based on epicness. So PCs who are focused around the same sort of saves (like bards) will find themselves good against some types of foes and poor against others. Classes with wide spell list have the meaningful choice of going for "the best" spells, or going for spells with a wide variety of saving throws even if they have overlap in effect (like AoE damage) to hunt for poor saves.
3. The three "strong" saves are the most commonly saved against, and include: The already most powerful ability score, the useful-for-everyone ability score, and another ability score that does happen to have the most common skill associated with it.
The first two points I find as valuable, the last one I find as counterproductive.
Moving to the 4e save (not "classic" saves) weakens #2, and strengthens #3. These are both moves in the wrong direction, so are arguably worse than the current system.
I have to disagree.
1. The 5e balance only holds true at low levels. Saving throw scaling is very bad, so saves cannot keep up with DCs.
At first level a typical PC will have 2 good saves it can pass with an 8 or so, another couple of decent saves where it gets a 50/50 shot, and one or two bad saves where a 13 or 14 is needed.
At level 20, the PC will usually still have two good saves (if they invested a lot in straight +2 ASIs, that is), and
maybe another decent 50/50 save if they also took the Resilient Focus feat, but all other saves will be terrible, needing a roll of 18, 19 or 20 to succeed.
2. 5e monsters are usually written to be complete trash at saves. Even most (supposedly) legendary monsters would be easily bypassed, if it weren't for their (IMO) inelegant but absolutely necessary Legendary Resistance feature. So, I disagree that 5e handles monster saves well: in fact, I believe it does very poorly.
3. Are you aware that the 4e save system allows for other scores beside Dex, Con and Wis to contribute to Reflex, Fort and Will? A 10 Dex, 20 Int character would sport a +5 to Reflex. If anything, this is exactly what's needed to move away from the sheer power of Dex, Con and Wis.