To your first paragraph....my players are slow. Yes I have to remind them all the time what statuses are on monsters. Hell they have to doublecheck their DCs all the time.It takes 30 seconds to realize that a foe that is prone is prone? Does the player not know that already when they want to attack with advantage? Do fighters with 4 attacks not usually repeatedly attack the same foe to take them out of combat rather than deal a status effect to several only for them to get straight back up again? Does it even take 10 seconds to roll the same saving throw for other foes when you’re expecting it because that’s what this PC seems to do repeatedly?
Make players responsible for their rules and I find they soon learn to remember. If you want to speed things up even more let them roll the saving through dice for the foe at the same time.
It is fair that once a fighter prones a target he should at least not have to remember saving throws, so that's a good counterpoint.
To your second paragraph, maybe I should. I am the classic DM mindset of "dm handles all that stuff"....but maybe it would be good for my party to do more of that, maybe it would help their memory
