Really? That seems a little unusual. The handful of places I've worked have all asked me what kind of direction I want to move in, with management path being an option (which I didn't pursue).

Actually getting those roles is another story. I've seen leaders go for many years without becoming managers, others switch over somewhat quick. But it seems like being open about what you want you work toward should, in a good work environment, get you some support toward it.

Doesn't have to mean you're trying to steal your boss's job. Could be stepping into a support or backup role for them, being ready to fill the shoes if they get promoted or job hop or retire, or could just mean getting some leadership practice. You can frame it as professional development, to be ready for other internal lead roles if they open. Then when/if they don't, use that experience to apply at other companies too.