Hi,
https://www.openstreetmap.org/way/1324330898 does not look like a riding centre; it looks more like a riding arena, similar to https://www.openstreetmap.org/way/1168323324. In OSM riding arenas are usually tagged leisure=pitch + sport=equestrian.

Also https://www.openstreetmap.org/way/499972019 is not a riding centre, it is a building, but because you have removed the tag building=yes it is no longer rendered as a building on the map.

The tag leisure=horse_riding should be used for the whole area of a riding centre and not for specific buildings, arenas or other things in a riding centre, examples: https://www.openstreetmap.org/way/1312659183 and https://www.openstreetmap.org/node/11837680455; compare https://wiki.openstreetmap.org/wiki/How_to_map_a#Riding_centre_(equestrian)