Fibonacci numbers are just a rounded version of phi^x. So the only coincidences are 1. that the number of teams is such that phi is a good base, and 2. that the rounding all happened to go the right way.
There's a fair bit of churn in these numbers: 51 clubs in total have been in the premier league in the period of interest (the last 33 years).
After some small threshold, I think the number of clubs doesn't matter. You could get the same result if the top 100 or all 40,000 clubs played in the same league every year, ignoring the minor scheduling problem this would cause. Resources are distributed approximately in a power law, as you suggest. What matters is the level of inequality near the top, which is apparently such that each team has approximately phi times the resources (measured over a long period) of the team below.
If you redo this table to quote PL first/second/third/fourth position per £ pound invested, (or total points in a season (3 for a win, 1 for a draw, 0 for a loss)) you get a different picture, e.g. for 2023-4 season: https://www.dailymail.co.uk/sport/football/article-13446423/...
Without rules, every club is going to invest huge sums betting on achieving a top result that would give return on investment, and only a few can achieve that each year.
Investors throwing lots of money at the sport is a feature, not a bug, from the viewpoint of the people involved in the sport. So by itself that doesn't need to be prevented.
Sadly aren't even succeeding at that given City's 115 charges (dating back to 2008!) and various clubs with their self-sponsorship shenanigans.
Unless you have enough oil money to just... not care about the rules.
Any theories why this might be the case?
> 2. that the rounding all happened to go the right way.
+1. See how close some title races are: 2013-2014 comes to mind.
The number of teams is constrained by "we need enough teams so there's actually variety" and "we can't have so many teams that we can't keep track of them all".