Are you going to go be able to go from Gulfstream to a major airline? Most likely not, they won't need pilots for a long time, and when they do there'll be so many more qualified pilots. I'd say go for a regional that you can grow into and that at worst case, could become your career. You can enjoy a lot more fun time instructing and some instruction gigs can pay some decent money these days, then you'll be competative for any regional you want, and while you might get turbine or turbine PIC at Gulfstream, realize that in the last 10 years, maybe longer, no one has worked at regional for a "year or two" and then went to a major airline.
I don't think that "greener" pasture exists if you're thinking of major airlines.
BUT, and this a big one, there are MANY opportunities out there, government, fractional and corporate (like you are discussing) and so on. Maybe if that's what you want to do, then go for it. Just don't commit yourself to the major-airline IMO.