9 League titles, Everton are a big club. They've also spent longer in the top flight than any other team in England I think (though Arsenal have the longest unbroken run).
Chelsea (dislikable or not) are now one of the biggest clubs in the world and even taking their paltry history into account they belong in the same category as Arsenal, Man Ure and Liverpool. The history argument here is an interesting one but if anybody thinks that Burnley or Huddersfield (for example)are bigger clubs than Chelsea they're just plain foolish.
Spurs and Man City have 4 league titles between them. They are spoofers of the highest order and don't deserve to be mentioned in the same breath as Everton or Villa (who at least won the league and European cup in the last 30 years).
Leeds are a big side for sure but their current plight has to be taken into account to say nothing of their incredible all time haul of 3 league titles (I think) and 1 FA Cup compared to Everton's 9 and 5 or Villa's 7 and 7 (no matter how dusty most of them are). Given that Villa and Man City have both experienced third tier football in the not too distant past Leeds fans shouldn't fear they too much. They can aspire to the heady heights of the upper mid table at best someday too.
Newcastle are over reliant on their gates too when they bang out about aspirations to join the big time. They've developed quite the losing mentality over the years.
Sunderland, btw, have no business on this list for the reasons outlined by LTID below. They have no meaningful history, have never even really established themselves as a premiership club (a few years decent showings under Peter Reid the closest they've come) and in effect are just a third rate Newcastle. List should look as follows:
Chelsea
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big gap
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Everton
Aston Villa
Newcastle United
Leeds United
Tottenham Hotspur
Manchester City
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another gap probably filled by a half dozen or so clubs...
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Sunderland.