Recommending a paper for rejection, then publishing my own

Is it ethical for me to submit my paper to a journal, citing the dissertation and giving it credit for introducing the problem?

Yes.

On the one hand, the material that inspired me to study this problem has been made public,

Yep, that sounds like a winning argument to me. If it’s public, then anyone may use it as inspiration for new research. Credit must be given to the source of course, as you yourself say you intend to do. The fact that you came to learn of the ideas through your work as a reviewer a year ago is irrelevant and I see no good argument in favor of disclosing it. The ideas are now public, period.

but on the other hand, I feel guilty for the amount of time that the authors spent on revisions that were ultimately fruitless,

Guilt is sometimes a worthy emotion, but I don’t see what role it has to play here. As the reviewer, you did your best to shepherd the paper along on its route to publication in the hope that it will reach the threshold of publishability, selflessly investing time and energy with no expectation of getting anything in return. Sadly, things didn’t work out that way, but it sounds like everyone involved acted in good faith and no one is to blame, least of all you. It would have been very different if you had developed the follow-up research while you were reviewing the paper and intentionally sabotaged its publication in order to publish your work first, but that’s clearly not what happened.

and this could lead to the original paper having a much more difficult path to eventual publication (somewhere)

That is too bad, but that is life. For you to voluntarily refrain from publishing a good scientific idea just to leave elbow room to other people who may or may not ever want/need/make use of it, would be damaging to yourself and (presumably) to the scientific community.

As for the idea of offering coauthorship to the authors of the original paper, that sounds like a reasonable option to me, but based on your account I don’t see that you have any ethical obligation to do so. In fact, depending on whether you feel they really deserve that type of credit, that may itself border on the unethical practice of gift authorship. Whatever you end up deciding, in my opinion coauthorship should not be offered as an act of charity or for irrelevant (even if true) reasons like “PhD students need to eat” as someone mentioned in a comment.

You should have recused yourself from reviewing the paper once you started working on the same topic if you had an intent to publish.

To not have done so represents a conflict of interest. How would you like it if someone did the same to you? Probably you would be outraged, particularly if the opposing paper appeared first and "scooped" yours. The publication of the thesis may make it "look" better, but not if the submission of the manuscript follows closely on its heels.

If you had done the work only to show that the existing manuscript was flawed—with no attempt of publishing your own work later on—that wouldn't have been a problem.

I'd reach out to the original authors, explain the situation and offer them a co-authorship. Once you have the manuscript ready to submit, that is.

Your post leaves unclear whether you would've thought of this problem and solution if you had not reviewed the paper initially. I think you would not have (otherwise I don't really see whether the ethical problem is, and you would have also recused yourself from reviewing the article), so it makes sense to give credit where credit is due. This holds for your method of solving as well - would you have come up with this if you had not read their solution first?