I have only dealt with airlines in the UK/EU (not sure if you are interested in this side of the industry). From my own experience and that of a few others I’ve spoken to, the airlines over here are absolutely and categorically not interested in interviewing non-EU/UK passport holders. They seem to systematically filter out candidates who don’t match their nationality criteria ie European, regardless of each candidate’s unique circumstances (type of visa held etc).
If you do have permanent residency however, it *might* work in your favour in some rare cases. I have heard of a non-EU pilot working for a business jet outfit on permanent residency. Perhaps carriers whose bases aren’t spread everywhere across the EU might consider you (in theory. If they really are equal opportunity employers).
Never heard of a company sponsoring a pilot’s visa.