Prediction should be a sport

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

Currently, people who don't want to gamble for money can take part in prediction making at https://www.gjopen.com and metaculus. Your alternative suggestion is to have all the prediction making happen in a shorter time frame during an event. A special event can bring people together, but most games of Poker don't happen in the setting of a tournament.

Poker has the advantage of being able to be played casually with low feedback circles. Adrenalin rises while you play and you don't have to wait a year to see whether or not you win.

I think we need games like the credence game that have tight feedback loops. One of the core problems of the credence game is that most of the questions aren't very interesting.

I can think of a few ways to get better questions:
① Get knowledge from Wikidata and make questions with the knowledge. Test every question in the database for being a question that increases player engagement and slowly optimize your way to highly engaging questions.
② Use real world data. Sensors like heart rate monitors provide interesting data that can be predicted. Weather API provide interesting data. Time tracking software can ask me question such as whether I spent more or less than 3 hours on facebook in the last week.
③ Tie prediction to common tasks with binary outcomes. It would be possible to ask for a prediction whether or not the test suite will find an error every time I run automated tests.

Of course finding ways to actually drive economic decisions with predictions like http://lesswrong.com/lw/oe0/predictionbased_medicine_pbm/ is also important.

In order to reach a level of entertainment as near as possible to the one provided by poker tournaments you should consider building a prediction game having at least two characteristics:

1) Restricted to low-stakes events (consequences of the outcomes are minimal and affect only a small set of parties; ideally within an artificial environment rather than real world events).

2) The timeframe should be reduced to a single day.

We essentially have this already occurring in the form of fantasy football leagues, which itself has gone from basically being gambling to basically being an e-sport. If you haven't considered it already, perhaps you should look into some of the ways that the NFL is making use of fantasy football for both marketing and information gathering purposes.

It's obvious - we need buzzfeed to create a "which celebrities will get divorced this year" quiz (with prizes?). There is no way people will be interested in predicting next year's GDP.

Yes, rationality could use more competitive tests of skill. I'm not sure about having them take a year, though. Have you thought about ways to make it faster?

There are way too many "shoulds" in this post. If anyone can have fun predicting important events at all, then it would probably be people in this forum. Can we make something like this happen? Would we actually want to participate? I'm not sure that I do.