I have to say that is exactly the opposite attitude to me and I would welcome them into the league.
One of the benefits of the pyramid system is that it rewards achievement, and if Bromley/Solihull or anyone else is good enough, then they should have the chance. After all, *we* had to 'apply' to be allowed into the football league initially - and the attitude that 'well they aren't an old school league club so we won't let them in' would have completely scuppered us! It's better now clubs get there on merit. I'd imagine there are fans of Prem clubs looking askance at the likes of Bournemouth or Brentford (both safe for another year) and saying they aren't 'proper' top flight clubs. If the 'proper old clubs' (who often have a bigger fan base than some of the smaller clubs) can't use their advantages well enough to get out of non-league then that's hard luck. One of them (Luton) have managed to resurrect themselves from non-league and get into the Prem within 10 years (much as I dislike that particular club).