I think lots of people achieve FI without something to jump to, both intentionally and unintentionally. Particularly if you mean a full-time interest, whether job or hobby. I think a very typical scenario are people who quit because they are burned out; they want to "do nothing" (in quotes, because more like just not work) to decompress, recover, and take the time to figure out what's next with a clear head and deep thought on their current circumstances, instead of where they were when they started their career.
For me, I had ideas, but ended up at FI 3 years early. I was laid off from my job in January 2020. A large severance (I was in Ireland at the time) mostly made up for the fact that I lost the job 3 years before I was going to quit. I came back to the US, and Covid gave me an excuse to try out retirement, while checking out the job scene without trying too hard. The stock market cemented things, and there is no need to worry about a j-o-b any more. I have already found expansions of my hobbies, and am quite busy with things I really like.
For you, you don't want a full-time product position. Would it make sense to take either a time-limited interim position, or a consulting role? The premise of the post is that you don't have a plan, and your question seems to point that you are concerned about that. I would think an ongoing role in one of your companies would give you a familiar base from which to really explore your next phase. While work will, of course, take time that familiar base could be useful as you think about your whole life, maybe for the first time or the first time in a long time. Or call it a sabbatical if you want, and try out something entirely new: being a novice at something can also be revealing on what you want.
The point is, it isn't the job or activity that's important. It's just the medium to think about yourself. And not having the burden of the whole company to think about may give you the space to think about yourself. Or, maybe It's still your baby, and you need to GTFO if you want to not go crazy with second-guessing the new leadership. If you are FI, you have earned yourself the luxury to just think. You might need to distract your body and your left brain with something in order to do that, but that need is entirely separated from the need for a paycheck now. Only you can decide what that set of circumstances is, and you might have to try a few out before you hit on one. Go easy on yourself in figuring that out.