One, there could always arise difficulties with sound being synchronous with the picture that is being broadcast just based on who turns on what at what time to get things going. It happens even with youtube videos in our wonder age of the 21st century (though those are errors on upload). And, as you see in those videos, a delay of even a half a second can, as the picture runs longer and longer, build up to the point where its a delay of several seconds to half a minute or more or anything in between.
That'd be good, but in those days the only people broadcasting were national groups, who presumably would have the resources to correct such problems pretty quickly.
Two, I'm not sure of the details of how signals work, but wouldn't terrestrial radio signals take some time to reach an area depending in its distance from the initial output? And wouldn't that also cause difficulty in syncing up the picture and sound?
Again, all EM signals move at the same speed, and you don't have to transmit the signals as separate either, since the TV is already dealing with a signal much more complex that mere sound could be, you can slot it in with few issues
Though I guess it's easier to just have a modernist totalitarian regime propaganda minister go all arty when he creates his official television format.
Arty can also mean having beautiful violin music alongside subtitled pictures if you want.
Having a TV without sound is like having a computer without a mouse, yeah, you could do it, but it would take a hell of a lot of explaining why.