Everton aren't a big club? They sell out their 40,000 ground every week, the Liverpool City Region has a population of 1.6 million, they haven't been relegated in about 70 years, and they attracted one of the most successful managers ever in Ancelotti.
They're an underperforming big club, but that doesn't take away from them being a big club. Them, Aston Villa, West Ham, Leeds and Newcastle are the second level of clubs below the big 6 for me (although I guess you could say Newcastle are making it a big 7 now).