alt Hacker NewsSouthwest and Alaska are absolutely major carriers. Not the absolute largest perhaps but major. Alaska is certainly not discount, and despite the name is not particularly regional.
To your edit, Alaska is something like 9.9%. Technically below 10%, but barely.
The Alaskan problem is they more often have unique routes. Which is great and serves cool stuff, but you can’t always count on it bringing competition to your main flight. Allegiant is similar.