Looking at the Mapy.cz website, they say in their FAQs that apart from Czech Republic and Slovakia, the rest of the world’s maps are all generated from openstreetmap.org data.
Yet when I go to Mapy, you can see trails marked with names etc on the outdoor map type:
But I can’t see to get anything similar to display on openstreetmap:
There are thousands of different map stiles out there, based on OSM data.
Maybe waymarkedtrails.org povides something close to what youe are looking for.
The map on openstreetmap.org is just one example, one out of many.
However, OP has a point: we have the richest database of trails and hiking routes out there, but fail to showcase them on the main website. We have two map styles dedicated to cycling in the main menu (Cycle Map and CyclOSM) but none for hiking, our arguably strongest point. Time to add one more, I suppose.
Something like Thunderforest Outdoors would be nice.