match frame rate should only change the video output when you start or stop playing, or as of iOS 14 if a screensaver kicks in when you've paused video.
The HDMI cable version isn't going to change anything, if you have an image at all, it's working. It's not going to slow down only parts of the video.
What apps is this happening in?
are you using airplay?
I'm guessing you're on an aTV-HD since you didn't mention range matching?