I disagree with this last. You're confusing the player stating what their character does with a specific game action. If I declare my character tries to knock their opponent prone, this isn't engaging a specific action, it's a description of what I want my character to do. The GM then engages with that, decides if it's uncertain or not (and if not narrates outcome), and, if uncertain, looks for an appropriate mechanical solution -- usually an ability check. And, indeed, we have rules here that help the GM in this middle part by suggesting that knocking prone is something that is usually uncertain and that it should be a STR check opposed by STR or DEX, what proficiencies are relevant for each check, and then constrains the outcome space a tad more tightly than usual (although not much in the knock prone case). This is just an additional bit of increased resolution to the rule details, it's not a special category separate from ability checks. Same with the spells or class abilities -- the player is declaring an action and pointing to the mechanic they'd like to use, the GM still evaluates, and choose to use that mechanic. Those mechanics constrain the outcome space, but not the process.
The using of ability checks to resolve an action is still the same process, just with a different set of constraints on the outcome space for the GM. If you'd like to talk about why spells are so specific on these constraints while ability checks are not, that's an interesting discussion to have, but I don't think it's particularly pertinent to this discussion.