So how do they know that they can use athletics with it? If I wanted people to roll strength check with a potential proficiency bonus from athletics I'd say "roll an athletics check." As athletics defaults to strength, that is the ability used unless otherwise specified and the mention of specific skill communicates that a proficiency bonus from it may be added. Simple and clear.
That's their call, and they should follow along with their declared action. If there's a question, we can discuss it, but I think most players can handle figuring out if athletics fits the bill. Further, I don't have to do the work of picking what skill to use, I just pick the ability. I already have a huge workload as GM, so anything that simplifies my life without causing problems is golden. Only having to pick from a menu of 6 things is a lot easier than remembering the whole skill list.
And, finally, I don't structure my games so that my notes say, "Athletics check, DC x." I assign both DCs and ability checks based on what action the players describe. Given they could decide to do anything, I don't have those notes. I might write, "west wall is crumbled and climbable, plenty of handholds but quite slippery with moss and water." That way I have a good picture in my head for both describing the scene and laying out the general options but also for fairly adjudicating action declarations. Someone that says they try to scramble up the west wall quickly, for instance, might get a hard DEX check because they're focusing on speed and grace with their action but it's going to run directly into the problem of slipperiness, whereas someone that says they're going to use daggers to wedge into the cracks and provide good holds might get an easy STR check because this approach avoids the slipperiness, leverages the many crags and handholds, but focuses on pulling yourself up with tools. Either approach could benefit from Athletics. The former could also attempt Acrobatics. I'll let my players tell me which they want to use, if any.
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In some instances asking for skill check is indeed sufficient to communicate intent. It depends on the context. But that is a separate matter. I could just as easily say that in some situations asking for an ability check with a proficiency bonus from a specific skill is sufficient to communicate intent. But when I earlier said that 'skill check is fine shorthand' I mean't exactly what I said: that it is an easy way to communicate that we are referring to an ability check containing a possible proficiency bonus from a skill, completely irrespective of who is using the expression.
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I disagree. A bald skill check involves the GM guessing what the player wants. I don't guess. Too many times that went wrong and I've learned my lesson here. It's trivial to expect a goal and approach, makes the game go faster because everyone's clear on both what's happening and what's at stake, and lets me hand more of the control over what happens to my players. If they describe an action that will clearly work, then it works, no need for a roll. If they ask for a skill check, well, they're betting on the dice when they don't have to. And, I'm guessing what they actually want to do, so the stakes aren't as clearly set.
I mean, I used to play exactly as you do, and I made the arguments you're making. Then I tried doing it differently, and, well, it works awesomely. Now, if I was playing a different version of D&D, then, sure, skill check asks are back on the table because that's how those games worked and how they were designed. 5e, though, is aimed slightly differently. Not enough that you can't use your method, but enough that I've found my method to be superior in outcomes -- for my table. As such, when I discuss how I play and approach 5e, I'm going to clearly state the process, which will be a bit wordier than what happens at the table. But, even in doing so here, you were confused how that play worked, so it's probably a good thing that I explain more clearly even than I have been.