Generally when you're "truly" senior, it's pretty trivial to come in and start working tickets ie bugs and features so long as it's the tech stack you're used too. You've seen most of the bugs before, and implemented a lot of the same stuff (dashboard? KPI's? yawn). If it's a different stack, then sure there's some ramp up, but weeks not months. Like if I went from 15 years of java to .net, I'd be uncomfortable for a while, but basics are the same; version control, IDE's, building, deploying, etc. It's just a matter of learning the particulars.
So far as projects to work on, generally managers and project/product owners have a back log of stuff they want done; some feature or other. Or maybe there's some crippling technical debt that needs to be resolved (example I saw recently; using ant for building deploying, and needing to move to maven).
>> If you want them to come up with their own project, at least intro them to the current strategic goals of the company and whom to talk to find and prioritize pain points.
Rereading; this is a bizarre statement. If I encountered a situation like this, I'd start looking immediately. This is a strong signal the org has no clue what they're doing ie leadership is incompetent.