You've got the makings of a good point in there somewhere but I'll help you get it out. There's no point using FA Cup finals as a benchmark because you don't need to be a huge club to get there. The last seven finals have had Wigan, Villa, Stoke, Hull, Portsmouth and Palace in it.
There's also no point looking at the top end of the Premier League. For about a decade, only Man Utd and Arsenal won it - and they were so far ahead it looked like they would dominate for another decade. Then Abramovich came along and now the top of the league is as open as I have ever known it to be.
What you need to be looking at is the bottom end of the PL. That's what can become a closed shop as the teams that get relegated have a strong chance to bounce back straight away and it becomes ever more difficult for other clubs to get promotion (as Leeds will testify). But the top of the league doesn't prove your point at all because there are half a dozen clubs who have a very realistic chance of winning it this season, and even the current champions aren't in that list. Next season it will be just as open.
Now you can thank me for helping your argument.