Maybe. I at least feel like there is a shortage of various types of prediction markets.

For example, prediction markets could be funny: How many beans are in this glass? You could run one during a live stream and see how the probabilities start to come closer to the truth the more beans you take out of the glass.

Prediction markets could be short-lived: Will Tom arrive on time? All his friends say no so they all would bet 99 sats to win 1 sat, but Tom thinks he will arrive on time so he can bet 1 sat to win 99 sats. If he is on time, the friends lose but they receive a Tom which arrived on time. Maybe that was worth 99 sats to them (kind of like hedging but against events, not on commodity prices). And Tom receives these 99 sats. Of course, you always have to account for perverse incentives since you don't want to incentivize people to be regularly late in the first place but I hope you get the idea. Just some fun with sats betting on random stuff which even exposes some valuable information possibly: Did Tom know that so many people are 99% sure he will be late? Also, the Lightning Network could make prediction markets very accessible: no KYC, instant settlement, micropayments and low fees.

And prediction markets could be user-generated and private (like the example above)

However, they are highly regulated (for example, to prevent assassination markets). I think that's why there are so few out there. But something about "bitcoin doesn't care".

You mean because of the financial incentives? Not necessarily if you use bitcoin, I think. But not sure I understand what you mean with "recreating the existing system".

Whenever you want to incentivize people to tell you the truth what they think will occur in the future.

I really like this video: Prediction markets: can betting be good for the world?

And this one from Robert Hanson. Hanson created a nice automated market maker for prediction markets.