Fortune/Misfortune Effects
Many affects allow the player to re-roll saves, either for good (Fortune effects) or ill (Misfortune effects). All of these would need to be revisited on a case by case basis to see how they would interact with your change. This includes Hero Points, as @brandon mentions, but also many important feats and abilities:
Most of the above can be pretty easily converted to instead make the enemy re-reroll appropriately, but there's all sorts of small things you'd need to watch out for - for example, does Incredible Luck grant the circumstance bonus to the player's save DC? Or do you apply it as a circumstance penalty to the enemy? (I recommend the former).
The Bigger Problem: Math
Again, this can be accounted for, but it's not as simple as just switching from the DC with +/- 10. It's a little counterintuitive, so here's an illustration: Bocephus the Goblin has a +10 Reflex Save, and Abraxias the Evil Wizard casts Fireball on him with a Spell DC of 20. Results for the normal rules (die roll, save result):
- Critical Failure (due to natural 1)
- Fail
- Fail
- Fail
- Fail
- Fail
- Fail
- Fail
- Fail
- Success
- Success
- Success
- Success
- Success
- Success
- Success
- Success
- Success
- Success
- Critical Success
This means there's: 5% Crit Fail, 40% Fail, 50% Success, 5% Critical Success
However, if you have Abraxias roll at +10 vs Bocephus's Reflex DC of 20, you get:
- Critical Success equivalent(due to natural 1)
- Success Equivalent
- Success
- Success
- Success
- Success
- Success
- Success
- Success
- Fail Equivalent
- Fail
- Fail
- Fail
- Fail
- Fail
- Fail
- Fail
- Fail
- Fail
- Critical Fail Equivalent
That means now Bocephus has a:
- 5% chance of critical failure (the same)
- 50% chance of failure equivalent (instead of 40%)
- 40% chance of success equivalent (instead of 50%)
- 5% chance of critical success (the same)
That's a huge swing in the favor of Abraxias. If you apply the same rules to players and monsters, it's at least seems equally penalizing to both sides, but many monsters already have a higher Spell Attack to compensate for this, so the players will still be extra penalized. If you want to make the math work, you could try adding 12 to Bocephus's Reflex Save to get his DC... but even then, as @Ruse points out, that alone doesn't solve it. Even using 12 instead of 10, Abraxias would critically fail (aka Bocephus would critically succeed) when he rolls a 2 or 1. So you'd have to make critical failures for this roll only happen when failing by more than 10. Again, not impossible, but I prefer to minimize complications at the table, and having two different DCs with different rules (one for normal things, like Demoralize, another for inverted saves like getting Fireballed) for each save seems awkward. (As a side note, getting to be the one rolling the die is an often overlooked advantage of Spell Attack spells.)
I've never met a player who doesn't want to roll dice
I've not played Pathfinder with everyone on planet Earth (yet), so I can't say for certain... but every player wants to roll dice. It's fun. I wouldn't recommend rulings that cause players to roll less dice, especially outside of their turn, when player engagement is at its most difficult. It's not really as fun to just watch the GM do everything as it is to get to roll dice in response to the GM doing things.