Yes, I know one can render a map from the database, and in fact I have been since 2014 ;)
Unfortunately the main site is the main reference point and is highly visible to end users. Otherwise there would be no point in having a separate "name" attribute that differs from the name:lang ones.

Well, ideally it should be up to the user to choose a language to display, but of course this is much harder when they are burned into PNGs.

It hasn't been like this for 15 years though - it was changed from "Snowdonia National Park" to the slashy format a month ago (changeset/84718316)

Well, osm.wiki/Wales states that "Welsh names can be tagged using name:cy=*" and that "An OSM rendering showing the Welsh names is available at https://openstreetmap.cymru/".

There is a longstanding convention that the place names are in English on the main OSM and in Welsh on https://openstreetmap.cymru/

Yes the heatmap is correct. The cycle path has been realigned as per the Apple Maps aerial imagery.