The problem with your argument is that the tool you are using to measure "great" clubs is entirely arbitrary. It's basically just your opinion. Which many in the media share. When Ancelotti took over PSG, I believe they hadn't won the title in 19 years. Mourinho just took over Spurs, they haven't won it since the 60's. Are they now great clubs? And the list goes on.
Great clubs have a history. Not just from the birth of the premier era but much longer, generation after generation in fact. By any reasonable assessment Everton are a great club. Indeed, It was one of the first things Ancelotti said in his conference today.
I see the point you're making but I think you've drank too much of the media hype kool-aid.
It’s not arbitary, as there is a clear system. It’s the only metric that truly matters when determining the size of a club, particularly in relation to the acquisition of top players, who ultimately care about competing at the highest level and playing with and against the very best players.
Consistently competing in the Champions League, where the best players are, and for major honours. The top six have all won league titles, domestic and / or European cups.
PSG are a big club because they win league titles, domestic cups and compete regularly in the Champions League. Spurs are a big club because, although they haven’t won a league title in 50 odd years, they’ve been top 4 and in the Champions League consistently for years and reached the final this year.
It often suits our agenda to say Everton are a big club and in our eyes that’s fine, but it doesn’t matter in any meaningful way. And naturally, Ancelotti is compelled to say we are a big club in his first presser, he’d be foolish not to. We are absolutely a great club, but in my opinion, we’re not amongst England’s elite, or Europe’s elite.