Postal stamps also show NIPPON, but in our daily conversation both nihon and nippon are used quite often.
When cheering on Japan at sports games, we do say "nippon!", most probably it is easier to stress/emphasize. "Ganbare, nippon" is a popular phrase.
And we do say "nippon jin" though "nippon go" is far less popular.
Languages (not only Japanese) are living things, and they can be illogical in many aspects.
(IMHO)