An average club can become an elite club, all this depends on the kind of coach hired for the job and the presence of outstanding players. Girona can survived European competition if they maintain consistency in their games.
A good coach can take an average club to a higher level but he won't be able to maintain that standard without proper funding. If the team isn't able to compete for the signature of good players then that team won't be at the top for long. They may be able to be at the top for one or two seasons. We've seen how teams like Valencia and Sevilla keep suffering because of financial problems.
Villarreal too is one of such clubs.
La Liga needs to reduce the strict financial regulations they have on La Liga teams. It's weakening the league.
Sevilla is a bigger club than Arsenal but Arsenal is more recognized because Sevilla doesn't have the money to attract big players.