Actually, as I understand it, that is already the case. Almost. But we can improve it somewhat to make it more effective.
Thoughts on the current policy
See, the way I see it is that a post that follows the homework policy really doesn't need the tag in the first place. My thoughts resonate with what you say:
I think this is because if you've put the effort in to identify the concept you're having difficulty with, it's then easy enough to take the "homework" part out of the question and just ask about the concept itself.
Instead of making the primary focus of the question the homework problem, it's easier to just ask for the concept and maybe include the problem to exhibit your thought process.
As I once said in chat,
IMO an on topic HW question is indistinguishable from a normal question when the "this is HW" part is excised.
I also think that the "show some effort" bit of the homework policy is much less important than the "ask about a specific concept", though it is more objectively determined:
It's not enough to just show your work and ask where you went wrong. If you just need someone to check your work, you can always seek out a friend, classmate, or teacher. As a rule of thumb, a good conceptual question should be useful even to someone who isn't looking at the problem you happen to be working on
The current state of questions
From what I see on this site, the majority of closed HW questions don't get improved. Some people add effort, but most of these still aren't conceptual, and arguably never will be as many are essentially asking for help with algebra or some such.
I think 1 in 30 HW questions get reopened, and that too usually because they become borderline (not because they get improved significantly). Some stats on this would be good, though.
You mentioned that you ignored that tag and it felt nicer. I'm pretty sure that the questions you missed seeing were eventually closed, at least most of them. It's just a matter of speed, and that's a different issue.
So, what can we improve?
Like I said at the top, this is almost already the case. Almost. We can still improve:
Currently, there still are some HW posts that get reopened because they added effort. We can try to be more strict when it
Besides this, from the policy POV, the semantics of the current policy is that "Homework is OK, just read this long post to learn all the caveats". Changing it to "Homework is not OK, but here is how you may be able to convert your HW question to a good one" may lead to some good results. Ditto with the close reason.
Basically, we consistently change the semantics of our hw policy across the site, and instead of telling users to improve their hw question (the wording means that it will most probably still stay an hw question), we tell them to convert it into a conceptual question (which brings home the point that the question needs a major change of POV).
I don't have a way to conclude this post at the moment, I may do so later after thinking about it.