Am I correct in my understanding the word 'Mine' next to two crossed picks on topo maps is an indication there has been a valuable mineral deposit discovery of which had a validity exam performed?
No. The symbol simply indicates a known mine. The evidence of a successful validity exam would be the certificate of determination itself. Many many mines will be found on topo maps that were never subject to exam or patent.
I'll mention most of them aren't precisely placed. While some are almost spot on, expect up to a 1500' or so margin of error (500' seems pretty regular). It all depends on what data source was used when the map was drawn. In the case of old shafts/adits, it's not unusual for them to just be retained each map revision with no one knowing if it was ever right to begin with. Your best bet is to try and track down a possible mine name and research it from there. Just don't be shocked when the original location notice is based off a "fixed" object like "the big pine tree at the top of the hill."