Thanks to DynamicGaming1060 for this fix. He put a Youtube video here showing this fix in much greater detail.
The codec that is supposed to enable Sony Movie Studio to play .mp4 files doesn’t work. DynamicGaming1060 says to replace compound.dll (I found mine in C:\Program Files\Sony\Movie Studio Platinum 12.0\FileIO Plug-Ins\compoundplug\) with qt7plug.dll. (You can do a Google search to find the file.
As usual, this is a “Do this at your own risk” item. Be aware that replacing DLLs can ruin a program. Verify the source of the file you download as safe.
All that being said, I did it and it worked. I can now work with .mp4 files in Sony Movie Studio.