Hi,
leisure=horse_riding is the wrong tag for riding arenas like https://www.openstreetmap.org/way/1370137400 and https://www.openstreetmap.org/way/1370140732 . In OSM riding arenas are usually tagged with leisure=picth + sport=equestrian.
leisure=horse_riding could possibly be a proper tag for 'pendor stables' (the whole faclility, not just one building). See also:
https://wiki.openstreetmap.org/wiki/How_to_map_a#Riding_centre_(equestrian)
https://www.openstreetmap.org/way/824942449#map=19/51.591787/-0.415199&layers=P (example 1)
https://www.openstreetmap.org/way/28992949#map=18/51.610148/-0.461855&layers=P (example 2)