"another local Okinawan sport, not only because it provided me with many hours of fun when I was young"

Anther, meaning separate,

When I was young, past tense probably not doing it any more, replaced it with karate.

I would think that if it was played/practiced as a youth and was part of his karate training he would have said that the wrestling he did as a youth helped greatly in the ground techniques that are found in his karate.

Actually, Funakoshi gives a very vivid detailed explanation of what Tegumi is. What you stated is only the first paragraph of his description. He even gives two specific techniques. And he does say certain bouts being with you laying down. Do you lay down standing up?

The Okinawan name for our style of wrestling is tegumi, and should you write the word, you would use
the same two Chinese characters that are used to write karate’s kumite, except that they are reversed.
Tegumi is, of course, a far simpler and more primitive sport than karate. In fact, there are few rules except
for certain prohibitions: the use of fists, for example, to strike an opponent, or the use of the feet and the
legs to kick him. Nor are opponents permitted to grab each other’s hair or pinch one another. Prohibited
also are the sword hand and the elbow blow used in karate.

Unlike most forms of wrestling, in which the participants are lightly clad, entrants in tegumi bouts remain
fully clothed. Further, there is no special ring; the bout may be held anywhere—inside the house or in
some nearby field. I should note that when I was young the outdoors was generally the scene for tegumi
bouts because they tend to get rather lively and our parents did not like to see sliding paper doors and
tatami mats damaged. Of course, when we held a bout in a field, we first had to remove all the rocks and
stones that are a prevailing feature of the Okinawan rural scene.

The bout begins, as sumo does, with the two opponents pushing against each other. Then, as it proceeds,
grappling and throwing techniques are used. One that I recall well was very similar to the ebigatama (leg
block and three-quarter Nelson) of today’s professional wrestling. When I watch wrestling on television
nowadays, I am often reminder of the tegumi of my Okinawan youth.

The referees were usually boys who acted also as seconds to the opponents, their principal role being to
ensure that neither participant was seriously injured or knocked unconscious. To stop the fight, all that
any boy who felt he had had enough needed to do was to pat his opponent’s body. Some boys, however,
were so dauntless that they would go on fighting until they were knocked out. In such cases, it would be
the duty of the referee to try to stop the bout before that happened.
Like every other Okinawan boy, I spent many happy hours engaging in or watching tegumi bouts, but it
was after I had taken up karate seriously that I came to realize that tegumi offers a unique opportunity for
training, in that it need not be limited to two participants. One (usually, of course, an older and stronger
boy) may take on two or three opponents or as many as he feels up to.

Such bouts begin with the lone wrestler lying down flat on his back, his opponents pinning down his arms
and legs. Once I had determined to become a karateka, I used to get four or five younger boys to wrestle
with me, believing that such bouts would strengthen my arm and leg muscles as well as those of the
stomach and the hips. I cannot now say how much tegumi actually contributed to my mastery of karate,
but I am certain that it helped fortify my will.