I am a little late to the party on this, but I am a bit surprised that no one has brought up all of the ways that a PC can cope with a high DC saving throw that they do not have proficiency in. 5e is designed for PCs to excel in dealing with some things while others things are a real danger. Having a low saving throw in some areas is part of what the game was designed around. While the high DC is difficult, there are so many ways of dealing with the high DCs of difficult creatures than just watching failed saving throws. These exist specifically to help PCs with the saving throws they struggle with.
A very partial list:
Barbarians have advantage on Dex
Bards give inspiration dice
Clerics cast bless
Fighters have reroll failed saving throws
Monks have proficiency in all saving throws
Paladins give a bonus to saving throws to everyone around them
Rogues get proficiency in Dex, Int, and Wis
Divination wizard sets a die to one of the 3 rolled at the beginning of the day.
Halflings have lucky
Lucky feat gives a re-roll
Resilient feat grants proficiency
Heroes Feast gives advantage on wisdom saving throws
Ability Score Increases can add a +1
Even then, a PC with up to a +0 in a saving throw after all of that will still fail against an Ancient Dragon's DC of 24, the Goristro's DC 21 saving throw or any number of other monsters with high DCs. However, the PCs still have other means of dealing with that aside from making a saving throw (working off these from memory so some might be slightly off). Another small list:
Barbarians have immunity to charm and resistance to damage while raging
Monks have immunity to poison and disease
Ancients Paladin gives resistance to damage from spells
Mind blank protects from charm
Forcecage, wall of force, banishment, resilient sphere, maze, etc. all provide a respite from a particularly dangerous monster
Dispel magic and counterspell stop spells that are causing problems for someone
Healing damage taken
Means of bringing PCs back to life.
Respectfully, the game provides so many ways to improve a low saving throw and mitigate the damage from a failed saving throw that the low saving throws that PCs having some saving throws they will never be able to make solely through a single d20 roll is a non-issue. The system does not break down against high saving throw DCs because the system is about more than that single d20 roll.
My party has been level 15 or higher for all of 2016. Next week they hit level 20. While most of the high DCs they have run up against are in the 19-20 range, that is still extremely difficult for someone with a +0 or +1. They have also fought and killed an ancient green dragon, an ancient blue dragon while fighting off 2 adult blue dragons, a fang dragon, and fought off an ancient red. They killed a lich with a robe of the magi. In the entire campaign, there has been one permanent PC death, and that came a month and a half ago against a creature with DC 18 saving throws when a PC failed his saving throw and got separated while the party dealt with something else.
Improving PC saving throws by scrunching them down to 3 options has several knock on effects
* It makes it easier for a PC to make some saving throws, but it does not ensure the originally stated goal of having no low saving throw bonuses. Certain class combinations can still have a +0 in places. Even a +3 from ability modifiers is basically useless against a DC 24 from an ancient dragon: Barbarians and Fighters can have low will saves; Bards and rogues can have low fort; clerics, druids, warlocks, and wizards can have low fort and reflex; paladins can have low reflex and fort; sorcerers can have low reflex.
* Because it is easier for a PC to min/max good saving throw bonuses in all saving throws (reflex, fort, will), it reduces the special nature of the monk's class feature to have proficiency in all saving throws.
* Because it is easier for a PC to min/max good saving throw bonuses in all saving throws (reflex, fort, will), the CR of any creature that primarily uses saving throws to attack is effectively reduced (not just the high-level ones).
* It takes away the specialization that saving throws generally have in this edition: STR to not be pushed around, DEX to dodge out of the way, CON to resist the effects against the body (poison, cold, etc.), INT to not be fooled/resist psionic, WIS to keep control of the mind, CHA to maintain wholeness of self (banishment)
* It causes odd interactions with class features. Evasion now works with a high reflex saving throw granted by a high strength score. The barbarian has advantage on reflex saving throws boosted by its strength score.
* It puts the spotlight on saving throws and ignores super high attack bonuses from high level monsters. It seems odd that low saving throw bonuses are addressed but low AC is not. A +17 to hit will almost never miss someone without magical items, heavy armor and a shield, or monk or barbarian.
tl;dr - This solution does not necessarily provide what I understand the desired result to be (no low saving throw bonuses) without min/max, reduces the danger of monsters that use saving throws but not those that use attack rolls, and has odd interactions with other parts of 5e.