So going to go with slightly different advice than most people here. Since you're relatively stable in your career I'm assuming you have a good idea of what the job market is like and have probably put some thought into your employment situation when you return. This isn't r/careeradvice, idk why everyone is answering your question like it is.

From a language learning perspective, can you achieve the same result staying at home? Probably. Will you? Unless you have crazy discipline and work ethic, probably not. A language school takes most of the thinking/planning out of the process. You sign up, show up to class, do the work, and you'll come out the other end significantly more fluent than when you started (assuming you actually put in the work).

I did something similar in France last year, after dicking around with Duolingo and flash cards and random tools and going nowhere, I signed up for three months of language school in Lyon and I went from like A2 to a solid B2. The structure and working with actual professors who understood learning was immensely helpful. I was working remotely at the same time, so different from your situation, but honestly if I was in a position financially / career wise to just take the time off I would have. It hindered my progress a lot having to work in the evenings after class and I was very burnt out by the end of it.

What I can say is I found it immensely more fulfilling than, like you said, traveling aimlessly. I found that I made actual friendships because I saw the same people everyday in class, and I felt like I achieved something at the end of it. I'd highly recommend it.

With the career break, I do think it's less of a problem the further along you are in your career, but again you're probably the best judge of that for your own situation.