When I started karate (1980), we would wrestle during warm-up. Usually sitting down back to back. At hajime, try to get top position and have control on opponent. very good warm-up. I think it came from judo.

Specifically instructed was fighting while one would be on the ground and one would be standing. Attacking hip/groin/knees/feet including take-down and submission techniques/locks. Goal was to get up as quickly as possible.
This made clear that fighting strategy in karate was standing, at least how we trained it. I have never seen it otherwise. I later saw a book on Fukien dog boxing, and recognised a lot of techniques.

We also studied 5 sanbon kumites wich would all end taking down the opponent, 2 of them fighting while both would be on the ground, but strategy would be to take out opponent by hitting or kicking and get up, although initiated by throw or lock or both.

I first heared of cage fighting in 1992.

Links to tegumi/ground wrestling.
Tegumi was a sport teached in the schools. Miyagi did it in highschool. It was a sport for commeners and keimochi. And there would be bouds during festivals. As of 1901, karate is teached in the highschools on Okinawa. I have no reference if tegumi was still teached at that point.
However many karateka also studied judo in the 1920-1940.
For those in law enforcement, it was mandatory.
I do not feel tegumi is incorporeted in karate. But it was studied as seperate art, later replaced by judo to some degree.