Okay. So it sounds like you are able to change the search name for the movies, and find a correct result. It also sounds like when you do, there is a Cover Art image associated with the information displayed. So MC should be saving that Cover Art. MC uses the site
https://www.themoviedb.org/ for Movie data. It wouldn't hurt to check on that site manually for one of the problem movies and see if it actually has Cover Art for it.
Also, as metadata for movies is saved in an XML file next to the movie file, it sounds like you have file and directory permissions set correctly to allow the writing of files to the folder the movie is in. The metadata, including the movie name, updates correctly so the XML file must be successfully saved to the movie folder. The Cover Art file will also be saved next to the movie file, so it shouldn't be Windows stopping the file from being saved. You should check in the movie folder to see which files have been saved next to the movie file.
As the movies you are having trouble with do have a [Name] tag equal to the file name, you could try changing the [Name] tag to the correct movie name before doing the lookup. That shouldn't make a difference though, and would be a bug if it does, since you are changing the search name and finding the correct movie information.
Certainly try what Brian suggested. But if that works, then it seems even more likely there is a bug in the standard "Get Movie & TV Info..." function. However, I am using the current public version of MC at the moment, MC21.0.66, and I used the "Get Movie & TV Info..." function yesterday, and it updated movie Cover Art correctly for the movie "The Hunger Games: Mockingjay - Part 2"
So, that leaves me with two ideas:
1. If you get the message "saving changes <number> remaining" then maybe there is a problem with the source file availability, so MC is waiting for the file to download. Or your internet is really slow. Most movie Cover Art files are under 1MB. Do you leave MC running to see if that message ever goes away, or do you close MC before it goes away?
2. Again, if that message shows up, it is possible that your virus scanner/security software is preventing the download of the Cover Art file. As the Cover Art is just a JPG file typically, the name of the file may be suspicious, or your security software may find something in the file that it thinks is a Trojan virus or similar. Check your security software to see if it blocked anything. Try downloading the full size Cover Art image from
https://www.themoviedb.org/ and see if you can save it (that may not be a valid test, but worth a try to see what happens). Turn off your security software temporarily and see if the "Get Movie & TV Info..." function works correctly.
See how you go with all that, and get back to us.