I can only frame this in terms of CA law. If there is a contract, even a verbal one, you have to give the contractor the opportunity to execute the contract before you fire him, or take matters into your own hands. That's what the demand letter is for. It starts the clock. Among other things, it states (1) what the original deal was (dates and scope, etc), (2) what's been done to date (right or wrong) and/or what the problem is, (3) what must the contractor do to make things right, (4) how long he has to do it, and (5) that you intend to (a) cancel the contract (fire him), and/or (b) hire someone else to complete the contract, if he doesn't perform by the deadline, and finally (6) that you will sue him for damages (your actual out-of-pocket expenses to make things right) if he doesn't perform by the deadline. In CA it's all of the above, but especially #6, that makes it an official demand letter.
Then you wait out the deadline (which must be reasonable). He'll either comply or not. If he complies (and probably even if he partially complies), you're kinda stuck with him. If he doesn't, then you're free to finish the pool yourself, or hire someone else to do it. And then you can try and sue him for your actual damages. You'll have to prove that the letter was sent, and that he didn't comply, and what your actual damages are. You won't get "pain and suffering" or punitive damages, etc. You may or may not win. You may or may not be able to collect the damages.
But if you just fire him, without the demand letter and the deadline, then, believe it or not, you're in the wrong and it makes it much harder to sue and win, if not impossible. And where it gets even more vague, is if you perform actions that undermine his execution of the contract, he could possibly use such actions to cancel the contract himself, which may or may not be a legitimate cause (to a judge).
So you have to be strategic when you pull the demand letter trigger. It might get him back on the job (whether you want that or not), and it might completely alienate him. Alienating him might mean you never see him again, or it might mean you're stuck with a contractor building your pool that despises you and might be willing to take it out on the quality of the work or materials. Of course, it's possible the alienating has already occurred, justifiably or no.
So be careful about "end-running" your contractor (or whatever your intention is for going directly to the supplier), because that could backfire. Just be prepared for the consequences: like ending up with a guy you don't want building your pool badly, or a guy you do want building your pool taking even longer and/or doing it badly. Or you could be jeopardizing a relationship with a guy that would have done a good job otherwise, but just had some supply-chain issues. Or you might just get a guy off his butt and get him to get started. Probably other scenarios, too...
So you'd google, or seek some small-claim advice, or advice from a contractor's license board, to see if Ohio works the same way, before you write a letter, and before you start taking matters into your own hands (however innocent those actions might seem to you).