I get the same behavior as you did, but I have a solution.
The file seems to be an automatic conversion from the source live feed. That may produce odd files for editing.
The file doesn’t seem locked or protected, as I could use a tool to import the file, then save as a new file. That new file was then easily editable.
The tool I used was Subler. The only thing I did was open in Subler, Add both streams (video+audio), then Save As with the settings as in the image:

I noticed that I had to import (Add streams, open as Untitled) like it was a non-native file format (never seen that for .mp4 before). So maybe the source was not a standards compliant .mp4?
The Save As restructures the file in a way that QuickTime (and other Apple apps) may like better. Nothing is re-encoded, so the quality stays the same as in the source.
- I gave it a new filename, by appending a version number, so to not overwrite existing files.
- The file format settings “MPEG-4 movie” saves it as .mp4.
- The option “64 bits chunk offset” is needed for files over 4GB (as this file is).
- The option “64 bits times” is only needed for files over 10 hours in duration.
- The option “Optimize” writes the file with easy-to-stream interleave, as well fast-start meta data at the beginning of the file (instead of the end of the file).
Once saved, QuickTime likes the new file much better, and will allow edits.