There are some corrections to your question.
It seems to me like a bad divorce between the 22 involved clubs and
the FA.
Nope, it was a divorce between the 22 clubs and the Football League. The Football League is different from the FA. In fact the foundation of the Premier League states
The five (big) clubs decided it was a good idea and decided to press
ahead with it, however the league would have no credibility without
the backing of The Football Association and so David Dein of Arsenal
F.C. held talks to see whether the FA were receptive to the idea. The
FA did not enjoy an amicable relationship with the Football League at
the time and considered it as a way to weaken the Football League's
position.
So now to the question:
Why did the FA agree to rename its divisions and that the highest
ranked league will become the second tier of English football?
As earlier, the FA pushed this idea. The web archive article says
But rumblings of discontent had started in 1991 with the FA's
unveiling of their 'Blueprint for Football' which put forward the idea
of a Super League of 18 clubs. The Football League strongly resisted
the proposals, but on June 14, 1991 16 First Division clubs signed a
document of intent to join the newly named Premier League. Eventually
all 22 First Division clubs tendered their resignation from the League
and, after months of financial wranglings and threats of a players'
strike, the PL became official on September 23. The FA agreed to
continue with three up and three down promotion and relegation, vowed
to reduce the Premier League to 20 and struck a record five-year
£304 million television deal with Sky and BBC.
So in this case it seems that the FA actually agreed with the Football League to allow promotions for 3 teams from the Championship rather than keep it a closed shop, in order that those teams get part of the earnings. Note in the Premier League, your final standings at the end of the year specify the % of the Sky TV kitty you take home.
So even though positions 18 - 20 are relegated, they will get a proportional share of the TV earnings. There is an incentive to be 18th, rather than 20th.
In each year, there is a chance for 3 teams from the Championship to get into the Premier League TV revenue. and 3 teams fall out of the Premier League too.
