I think the real answer to the question in the title is "why not"?
The very first European Football Championship started in 1960, where 4 teams took place. This became a tradition and the next four editions followed the same principle. Then number of teams increased to 8 in 1980, to 16 in 1996 and finally to 24 in 2016. The number of teams in the final competition was gradually increasing by time as interest to European football was growing.
UEFA Euro was always growing. It started with 17 teams in the qualifying phase of 1960 edition, whereas now UEFA has 55 associations eager to get to Euros.
Another important reason for not expanding immediately, apart from interest, financial reasons or the number of independent countries in Europe, was that European Championship first had to become a brand. Football is this popular primarily thanks to clubs. They grow players, they pay them, they buy them, they sell them, they own them (it's true). In order to convince clubs to release their players (where some of them will return with injuries), especially for the qualifiers which take place during the season, UEFA had to make UEFA Euro a brand which no footballer and no fan could refuse.
Spain withdrew in the second round of 1960 qualifiers when was drawn against Soviets. The same thing was done by Greece when they were drawn against Albania in 1964 qualifiers. I can't imagine a similar thing happening nowadays, when European Championship became one of the most prestigious sporting events in the world. Azerbaijan and Armenia (which currently have no diplomatic relations between each other) are now in the same league of the UEFA Nations League 2018–19. Being highest rated teams in their groups, there are high chances that the sides will meet each other in the knock-out rounds. As someone from Azerbaijan, I can't imagine if one of the countries would ever withdraw from such a match, where a win will guarantee a slot in the UEFA Euro 2020.
The claim that FIFA may not wanted to give another international football tournament (which would be under the auspices of UEFA) a prominent role, is actually wrong, because UEFA associates itself with FIFA. Here is an excerpt from About UEFA page:
It is a continental confederation of the world football governing
And this cooperation dates back to the foundation of UEFA. Here is an excerpt from UEFA history - Early days and constant expansion:
[...] the movement supporting a body uniting Europe's national
football associations gathered pace after FIFA had approved the
statutory basis for the creation of continental football
confederations in 1953.