Most monsters & NPCs use the listed saves in the MM. I do give a few special NPCs all good saves. For a Conanesque sword & sorcery game like this one I'm ok with martial characters being stronger in
direct combat, I'm giving the lone caster (Warlock) the ability to use scrolls from any caster class and I would
let him do rituals - although I didn't let him summon his Archfey patron at 3rd level the way he wanted.
I agree the low-level game is balanced as-is, long as the GM keeps DCs low and doesn't run Caverns off Thracia

; I'm unconvinced high level 5e is balanced when PCs always fail most saves, but maybe I'm wrong. Would be good to see some accounts.
I can give you an account up to level 16 as to what you need to be concerned with.
1. The paladin: The paladin has an aura that gives Charisma bonus to saves. Combining the aura with proficiency in all saves might create a situation where players are making saves too easily. Prof. bonus + cha. paladin bonus + ability bonus can lead to a very high save.
2. Spells:
bless provides +1d4 to all saves. This one spell is the most powerful group buff in the game. If you don't have a class that provides
bless, then you don't have to worry about it as much. It's a haves versus have nots situation. That's how powerful
bless is.
Aura of Power: This ability gives advantage on saves against magical effects and spells. Combine it with the other options, you're looking at making magical effects and spells trivial.
There are other spells that provide damage resistance against many common attacks.
3. Class Abilities:
Fighter: Fighter has an ability that allows him to reroll a save a few times.
Barbarian: Barbarian has an ability that gives him advantage on Dex. saves all the time. His rage ability causes half-damage against nearly everything if he takes Bear Resistance.
Wizard:
Counterspell can often be used to counter dangerous spells.
Dispel Magic can remove a lot of negative spell effects.
Cleric: Cleric has quite a few spells for removing effects.
Paladin: Can remove spells by touch. Immune to frightened. Can remove disease and other effects with Lay on Hands.
Monk: Immune to poison and disease. Can spend a ki point to reroll failed save.
4. Monster Abilities: Most monster abilities are short duration or round by round saves save for a few very powerful abilities. Not a lot of AoE powers other than dragon breath and occasional AoE lair effects or spells that do a little damage. DCs seem to average between 17 and 21 for higher level creatures in the CR 18 to 20 range.
Green dragon lair action that charms.
Lich touch.
Most spells by casters.
Even random round by round rolls often lead to a high enough roll to break free from effects quickly, especially when buffed.
A lot of this is left up to your design. You might need to design tougher monsters at higher level or their special abilities won't have much of an effect.
At low level saves seem harsh and they can be. At high level it's not nearly as bad with all the character capabilities for protecting themselves. Just some things to watch out for when tailoring your game as far as saves go.