Speaking as an Expat......and having lived in various countries.......
Its really sucks for the up and coming pilots. (from any country)
Because with experience and a few type ratings (and flight hrs on those type ratings) you can really pick and choose where you want to work.
A Visa in just about any country is easier with a sponsor. (USA included)
Without a business sponsor....its damn near improbable.
With 250 - 500 hrs.....no one wants to give you a chance much less a sponsorship
Sad fact is Indian companies are sponsoring expats.
The Indian companies have convinced cajoled or cashed the appropriate people and the government issues work visas.
If the situation in Europe or the USA were reversed.....there would be companies sponsoring visas in both places.