You could have the prediction market invest the bet in a portfolio of the user's choice (maybe out of a few preset options) and then return the money plus investment returns (maybe minus a small fee).
no one wants to lock up $1 of capital to extract $ 0.10 of profit in a year,
That is an incredible inflation adjusted return. You need to lock up capital (sacrifice liquidity) to reduce counterparty risk to zero. Otherwise your bet isn't worth the value on the tin, as any rational trading partner must also consider counterparty risk discounting (which tends - not coincidentally - to be similar to inflation-free interest rates). This comes up all the time in crypto mechanism design, and in general it's part of the reason why ethereum has permanent inflation (as you may need a permanent income stream to pay stakers who are sacrificing liquidity for the positive externality of securing the network).
Have prediction markets which pay $100 per share, but only pay out 1% of the time, chosen randomly. If the 1% case that happens, then also implement the policy under consideration.
I don't think the capital being locked up is such a big issue. You can just invest everyone's money in bonds, and then pay the winner their normal return multiplied by the return of the bonds.
A bigger issue is that you seem to only be describing conditional prediction markets, rather than ones that truly estimate causal quantities, like P(outcome|do(event)). To see this, note that the economy will go down IF Biden is elected, whereas it is not decreased much by causing Biden to be elected. The issue is that economic performance causes Biden to be unpopular to a much greater extent than Biden shapes the economy. To eliminate confounders, you need to randomiser the action (the choice of president), or deploy careful causal identification startegies (such as careful regression discontinuity analysis, or controlling for certain variables, given knowledge of the causal structure of the data generating process). I discuss this a little more here.
Not an alternative, but an add-on: subsidize the market. This does require someone who wants the info badly enough to pay the subsidy. Robin Hanson recently blogged about how one might focus subsidy on trades one is most interested in.
P(A|B) is defined as P(A & B) / P(B), and both P(A & B) and P(B) are straightforward things to bet on in a prediction market.
The problem is that you get some estimates P*(A&B) and P*(B), and P*(A&B)/P*(B) is not necessarily a good estimate for P(A&B)/P(B) even when each of the component estimates were good. It gets much worse when the estimates aren't very good.
It gets worse still if what you really want is something more structured than a simple conditional probability, such as a credence for a causal relation. I suspect that there are many complications here that may be beyond the scope of any plausible prediction market structure.
Consider a conditional prediction market, e.g. "if my cool policy is implemented, then widget production will increase by at least 15%". To my understanding, markets like this are intended as a tool for finding P(outcome|event) and the market just gets unwound or undone or refunded if event doesn't occur.
I can work through the math and see that refunding the market indeed makes the price reflect P(outcome|event), but this exacerbates one of the biggest issues with prediction markets: no one wants to lock up $1 of capital to extract $0.10 of profit in a year, so no one will lock up $1 of capital to extract $0.10 of profit in a year and only if some extra event happens.
My question is: are there any interesting or viable alternative ways to run a counterfactual or conditional prediction market? Off the top of my head, I could imagine using markets for P(event) and P(event→outcome) to derive P(outcome|event), which would still pay out something if event didn't occur.