Something I did not encounter often is the fact, that GOM Player has some magic trick up its sleeve. It mostly allows to rename the currently playing files and sometimes I can even move them. I cannot really tell how that works, but it does (playing from UNC network paths here (ntfs, windows)).
Recent version got a bit bulkier than they were some years ago. If things get to much, use one of the older versions, they play mp4 and mkv just fine. Also needs some firewall tweaking to get rid of occasional ads in the free version.
This is the one player I've found that has extremely broad codec and container support, a reasonable amount of settings to manage, and above all else, fast and crisp thumbnail seek previews. I can do without all the hotkeys and skins other players offer to get those needs met in abundance.