Just beracause he did not mention tegumi being an integral part of the ground fighting in Karate, does not prove that there is no ground fighting in Karate.

There is in fact ground fighting in atleast some styles of Karate. If you do not have a teacher than can show you the applications you need to either closely examine the movements of your kata and get creative or just find a better teacher.

I have had 2 Goju Ryu teachers that showed me ground fighting applications. I had this discussion with another teacher who asked me if I know the ground applications from Naihanchi. One is an escape from the "mount"

Despite popular opinion, the Gracie family did not invent ground fighting. It's been around for a long long time.

Tegumi most certainly had a strong influence on the development of Karate and Karate includes ground fighting and stand-up grappling. Funakoshi may not have taught it in his "school boy" version of Karate, but that does not mean that it never existed.