One is testing performance of code (SQL or otherwise) and ensuring it achieves a particular level of performance or ensuring no regressions. This is a problem not just with automatically unit-testing SQL, but any other code. In general, for regular code, we simply go with special, "manual" tests of the execution time for critical pieces (the biggest issue is fragility or flakiness: due to changing conditions a test runs under, speed is not always stable). We could do similar for testing SQL performance. If we know the database we are targetting, we could also use some of the introspection tools it offers to get even better tests (eg. we could run an "EXPLAIN (FORMAT JSON)" query on a very large Postgres database matching production to ensure right indexes are being hit and no seqscans are being done and Postgres' estimation of the time is on target).
Basically, so far it's hard because "performance" means so many different things, but I don't think it's impossible.
As for the other point, those 1-in-a-million edge cases, we've got those with regular code too! If you really want to test something against production-like DB, it's not hard (make an anonymised replica of prod DB), it's just expensive (tests will be slow, getting this set up will be slow, etc).
I personally believe the right balance for cases like those is to catch them in production: automated tests should be quick and allow quick iteration if someone wants to do TDD on any part of the codebase. There are certainly product niches where this is not true (let's not have airbags in cars deploy accidentally every 1M rides, because there's a lot more than 1M rides daily :)), but for our regular applications, that's usually more than fine: quick tests will make it easy to fix the particular edge case once we hit it.
FWIW, I love the idea of pgTAP for those who haven't seen it too.
And that then extends to why staging and even dev aren't very useful - the scale is part of the equation. Even just doing your dev loop, it's pretty easy to get something that's logically correct on your test sample but then when you run across the real dataset you find that 1 row out of every million is weird but you still gotta deal with it.