Like Jet Blue, Alaska has a handful of wide bodies (that it got from its Hawaiian acquisition) and does a few international routes on its own metal, but it is an airline you fly domestically and that primarily serves the west coast most effectively.
It’s not a bad airline (it’s also not anything special), but it’s not the equivalent of American, United, or Delta. And that isn’t to say the individual flying experience might not be the same as those carriers, but it’s not the same class of operation as a business. Again, it does a handful of international flights. To me, major means you have an array of wide-body jets and a large international operation. Not a few seasonal routes on its 29 wide-bodies. For comparison, American has a fleet of 137 wide body jets. Delta has 179. United has over 240.
These are not the same.
Southwest is even less of a major carrier. It’s got a lot of marketshare from its previous era focused on selling the idea that it was cheaper than the majors, but it operates one type of jet, has zero wide bodies and its international destinations don’t count.
And while Alaska might have to use American to pad out its network of flights at major hubs and markets, at least it can. Southwest’s design to eschew major hubs means that tou can’t get a lot of direct flights to major cities. If it happens to fly where you travel to, fine, but otherwise you have to look at at least one hop and go to a smaller airport than what you might expect.
Jetblue doesn’t have widebody aircraft, but they still offer transatlantic connections.
If you're already on the east coast and aren't going to the west coast, then Alaska doesn't have a route. So I suppose they are "regional", where the region is about 70% of the country rather than all of it.
So yes, still a regional airline if your home base isn't exactly on the west coast.