Is it okay to work in a very old research area (in TCS)?

You have discovered a fundamental trade-off. If you work in an obscure field and do something significant it may be harder to get recognized, but you have the field to yourself and a few colleagues. But if you work in a popular area you might just get lost in the crowd and even get scooped in your latest findings. Neither is ideal. But either can be a path to success.

However, you need a base from which to work. But so does everyone. In looking for an academic job you need to match yourself to requirements and also show that you have the potential to stand out, at least a bit. If you are competing with 40 other people just like you in background and research interests it will be difficult to stand out. But if you are unique you will raise questions about how you will fit in. It is, however, your case to make.

It is easier, of course to do what you want when you are already well established (tenured). But you have to do what it takes to get tenured. But I don't think you need to defer doing what you like. You just need to do it well.

Personally, I find your interests intriguing. But that is because I studied math but worked in CS. I'm certainly not unique in that. Being able to collaborate across departments might be seen as a plus in some institutions. So might joint appointments between two departments. Everything is a tradeoff.

But spending your working life on things that don't much interest you just because they are "hot" seems like a waste of talent. Since you don't seem to be one of those fighting to get into one of the top ten institutes in the world, but can, perhaps, have somewhat more modest goals (top 50?, 100?) I think you can do fine.

So, yes, you can be a successful researcher, but you will have to make that success yourself.

I personally know many people (whose names I cannot disclose before consulting them) who work in the field of theoretical computer science and deal with pretty "old fashioned" problems.

However, each of them has a common characteristic: before purusing the research area that lies in their heart, they firtst published numerous articles in so called hot topics.

So, I believe that you can be a truly significant researcher in the area of your interest provided that you keep your current research as a side project and not pose it as your main area of ineterst while applying to jobs, fundings, etc.

I would also remark that (again, according to my own experience) people who stick to old-school problems and do not dwell into the hot topics are above-the-avarage scientists.