AOO is triggered by goblin. Game pauses. Combat log pops open and tells you the goblin triggered it. Your melee attack lights up. You click on enemy to use your reaction to attack or you just hit space bar to continue the round and not use your reaction. No popups. Just game pausing and if you want to know why, the combat log says why.
Enemy casts spell. Game pauses. Your counterspell lights up. Don't want to use it. Space bar.
This is mechanically the exact same thing as a pop-up, just in a different UI. So yeah, sure this way would be fine.
Ok. What about a combination of the ideas?
Uncanny Dodge could have presets, for example. If enemy deals x amount of damage or more, trigger Uncanny Dodge. If last enemy in the round hits rogue, Uncanny Dodge triggers regardless because it's the last enemy.
However, like Ragnarok said, you also have the option to set it to pop-up whenever you want. Maybe you're in a fight that requires more player control. So turn on popups for Uncanny Dodge. But if just fighting against a bunch of grunts, turn on presets.
Do this for all reactions. Have some basic presets but allow to turn on popups if needed.
Oh look, we've cycled back to the optimum solution of 3 reaction toggles: always ask me / automatic usage / off. Just with a more complicated "automatic" implementation.