I think your starting assumptions are false.
prediction markets ... aren't really catching on
Not true. Most financial markets are prediction markets. They seem to be popular.
stock market, which for the vast majority of people is no less distasteful
Not true. In fact, I do not know a single person who would characterize the stock market as "distasteful". Caveat: I don't know many tankies.
Only a small minority of people are neither disgusted by nor terrified of gambling.
Not true. Look at how many people are playing the lotteries, going to casinos, etc.
In general, "prediction market as a sport" is called trading the financial markets. HUGE prizes :-D
Not true. Most financial markets are prediction markets. They seem to be popular.
That may be technically true, but only in a superficial sense. Stocks prices have a very complicated relationship to real world events, except in the very long term. That's very different from markets like https://www.predictit.org which have clear connections to things like who will win elections and objective criteria.
So, I've been thinking about prediction markets and why they aren't really catching on as much as I think they should.
My suspicion is that (beside Robin Hanson's signaling explanation, and the amount of work it takes to get to the large numbers of predictors where the quality of results becomes interesting) the basic problem of prediction markets is that they look and feel like gambling. Or at best like the stock market, which for the vast majority of people is no less distasteful.
Only a small minority of people are neither disgusted by nor terrified of gambling. Prediction markets right now are restricted to this small minority.
Poker used to have the same problem.
But over the last few decades Poker players have established that Poker is (also) a sport. They kept repeating that winning isn't purely a matter of luck, they acquired the various trappings of tournaments and leagues, they developed a culture of admiration for the most skillful players that pays in prestige rather than only money and makes it customary for everyone involved to show their names and faces. For Poker, this has worked really well. There are much more Poker players, more really smart people are deciding to get into Poker and I assume the art of game probably improved as well.
So we should consider re-framing prediction the same way.
The calibration game already does this to a degree, but sport needs competition, so results need to be comparable, so everyone needs to make predictions on the same events. You'd need something like standard cards of events that players place their predictions on.
Here's a fantasy of what it could look like.