Not necessarily, because it depends which city your hub is in. Other airlines might have issues connecting to that city, but if their hub is in a city that doesn't have heavy anti-vax presence, then they are only slightly impacted.

Jacksonville FL, for example is a major SW hub, and was completely shut down. Similar issues exist in two other major SW hubs - Houston and Phoenix.

Also, not all other airlines have mandated vaccinations. ...and those that have aren't in heavily anti-vax cities.