The expression index will make it fast to retrieve the rows for which that predicate is true, but it won't help the planner know that this will be the case for 50% of rows, so I don't think it will change the join that the planner selects (which is the problem here).

In fact, this might make the query slower. If postgres thinks it is selecting a very small number of rows, it will prefer an index scan of some kind, but a full table scan will be faster if it's retrieving 1/8th of the table (at least, for small rows like these). So, you might get a slower row retrieval and the same explosively slow join.