You need to force a refresh. Usually shift plus the refresh button does this on browsers running on Windows; I think it's option + refresh on Mac but don't quote me on that.

I find it odd that OSM does not mark the image tiles as 'different' so people automatically get the new ones instead of the cached ones.

(Thanks for the ideas so far, by the way!)

Unfortunately, in both cases it would seem the road still renders on the map but marks it as "abandoned". The road is totally gone and shouldn't appear on the main map in any way - I'm trying to figure out how to get this addressed.