Here's how I understand it, though I could easily be mistaken.
The chronometer replacement was a close-up shot done completely in CGI. Not just the device itself, but the helm housing and everything else. If you saw a hand nearby that was CGI too.
My understanding is that CGI was mostly used only when the entire shot could be replaced (i.e. no live actors in the scene). Exterior shots, matte painting replacements, close-ups, etc.
It was a more difficult and expense process to insert bits & pieces of CGI into the original footage with the live actors and so forth, so it was rarely used.
I think in most cases, any given frame is either 100% CGI or 100% original remastered footage, with only a few exceptions here and there.
Like the example that started this thread for instance. All the actors in the Cage scene were CGI apparently, not a case of inserting CGI elements into a live shot.
I'm not saying it can't be done...Luca's did that Jabba The Hut scene for Star Wars...but probably wasn't in their budget to use very often. Especially since those Bridge monitors are in every episode.
Again I could be wrong, but that's how I understand why some things got replaced and others didn't. If it was original footage they generally left it alone, except for the remastering of course.
Not sure if anyone has more detail on this aspect or not.
Mark