There is absolutely nothing against the CPU cooler, just consider the height (15.94cm) and the space for the RAM (depending on the RAM height, it may be tight if all slots are occupied, but the fan can be mounted a little higher).
Alternatively, a water cooling system (already available from 80.-) would be an option, as the Ryzen 7 5800x is a hothead.
Some examples (I use the Artic Liquid Freeze in the 360 variant for the Ryzen 9, for the price tiptop):

Arctic Arctic Liquid Freezer II 240 (6.30cm)

be quiet! Pure Loop (5.20cm)

You can also get the Dark Rock Pro 4 for less than 60.- (note the size):
be quiet! Dark Rock Pro 4 (16.28cm)