The problem with 5e saves is that you'll have bad saves, at least 2. And when saves are your defense against spells, look at the problem in combat: casters have options, it is easy to pick 3 cantrips that require one save each and the chances you'll have one that is almost certain hit is high. Compare it to a simple attack that always aims AC, and everyone is trying to have high AC, this is unbalanced.
Bad saves always existed, but in 5e they not only increased their quantity but also gave spellcasters spells they can use at-will just like a fighter swinging his sword, and doing the same damage + some special effects!
3e had no at-will spells so the benefit for aiming a lower defense was balanced by the fact that you had to bur a spell slot.
4e also had bad "saves" (called defenses), but at least they gave the martials options to aim Fort/Refl/Will somehow. Not the best solution, but way more balanced.
IMO, more than how many saves the game should have, they should be balanced.