"as a DM, typically, I'd simply ask them to make those checks _before_ I'm starting to describe what they see!"
Then we agree on this much: the DM, not the player, is the one who decides when an Ability Check determines success/failure, decides which stat, and decides which skill proficiency, if any, modifies the stat check. (Wis +Arcana to check for a visible shadow, vs. Int + Arcana to be reminded that sometimes invisible things can move visible things...)
There are, of course, times when a player can reasonably offer a roll result. "I sneak up behind him. Stealth roll yields net 18." I might not *require* it - for example, when the player is actually sneaking up on an illusion or a statue, so their stealthiness doesn't matter - but I don't mind the *offer*, and if they rolled without need and got a 1, well, that's an opportunity to complicate the story, if it needs a bit of complication.
"if I'm activating a particular class or racial ability, it would only be polite for me to mention why I'm activating it, and what I'm hoping to achieve."
Okay, fair point. "I activate Divine Sense, scanning for any infernal presence", is a reminder if needed, a statement of intent, a nuance of your character's allocation of their attention, and *not* a back-seat-driver move. You know what you're hoping to achieve, and you've communicated that *intent*.
Umbran, interesting point re Inspiration when the DM makes a roll, such as a players (Wis + Per) Spot check. I think there's ways to allow player choice about Inspiration use, without the straw-man extreme of "I'm rolling to see whether you notice that you're being followed, so are you spending Inspiration?".