I probably know ten people who got hired at the Big Six with busts... one or two busts. Everybody I can think of with more than two busts is at a regional or LCC. I'm sure there are some at the legacies for whatever reason, but I've definitely observed that those getting the calls have either pretty clean records or insider connections (even connections are not a guarantee in that case, I can think of two guys with connections who expected to be shoe-ins to DL and UA, but their training/flying records got in the way).

I have to stick to my guns that if you have 3+ busts you might want to carefully consider taking an available LCC opportunity rather than holding out for the top tier. Would suck to hold out for five years and then end up at the LCC anyway.

With that said, I suspect the top tier will get to the point in the shortage where they can no longer hold irrelevant busts from your distant GA past against an otherwise qualified professional turbine pilot. But if you have more recent busts, especially in 121, they will be less forgiving of those if there is a pattern.

If you have to hook rides, better if it's entry-level GA, and better still if it's CFI rides.