As said at [https://help.openstreetmap.org/questions/60223][1], another way to get the tile URL is, with Firefox, to access the page information (which may be in the browser Tools menu or in the non-Javascript right-click menu, which you can open with Ctrl-Shift-right click).
The page information contains a Media tab with the URL of all images. It's not very handy, but it's feasible.
Of course it would be more sensible to have a "Request rendering" option somewhere.
However the best way may be to use the latest version of JOSM, where you may see the tile status or force rendering (right-click on the map).
[1]: https://help.openstreetmap.org/questions/60223