Yes, this should definitely be done for various reasons:
There is precedent for this on other SE sites, from Science Fiction & Fantasy, over Unix & Linux, to Gardening & Landscaping.
We are a classic case of an "&-site" like the ones above in that our site name contains two different but related and equally well-served topics.
The topic of TV is distinct and significant enough that people could very well be primarily interested in that (especially since the general cultural advent of TV-shows after the millenium), nor does the term "Movies" immediately suggest serving TV-shows, too. Yet it is a topic that is equally well-served by this site and deserves that recognition.
It's highly unlikely that there will ever be an SE site dedicated to TV (and bearing that title) other than this one. While there could still be future SE sites for topics that this site doesn't serve, like technological home cinema considerations, I'd doubt those potential SE sites would name themselves TV.SE.
It seems a rather easy and unobtrusive thing to do and I can't really come up with any disadvantages to the technical side of it.
There could be a few counter arguments, though, and I'll try to adress them:
People are already finding this site with their TV-related questions and the site is often found by other means than users randomly typing movies/tv.stackexchange.com into their browser adress line to see what happens. So the impact of this change could likely be cosmetic in nature. However, as said, that redirect doesn't really seem to hurt anyone either and we don't really know if people acquainted with SE actually tried that and failed. Nor is the mere cosmetic purpose of this change to be ignored either, as explained above.
A bigger counter argument could be that the term "TV" might be a little fuzzy, seeing how TV encompasses many things apart from TV-shows and that only the connection with "Movies" clears up that matter. However, the same could as well be suggested for "Movies", which encompasses many topics we don't serve here, like practical guidance on producing them. Yet, we rarely ever have people landing here with such questions already for Movies and there's always a certain degree of ambiguity with a site name before looking at the actual questions anyway. Besides that we do cover a lot of topics around the distribution and production of TV content (and not necessarily fictional TV-shows only), even if not all topics of course.
As pointed out in the comments, if this is followed through, it should probably also include television.stackexchange.com as a redirect, similar to the existing link redirection of sciencefiction.stackexchange.com to scifi.stackexchange.com.
