Here's a picture of an empty studio at NHK in Shubuya area of Tokyo. Lights look similar to "rubber chicken" game show studio lights. (but then again, all TV studios may look the same).
Maybe they fly into Tokyo, do the game show first, then take Shinkansen to Kyoto/Osaka area. Or maybe not.
Your hypothesis is reasonable and that may be what teams did in leg 11, but it's 250 miles from Tokyo to Osaka so I have big questions that the TV game show was filmed in Tokyo.
You inspired me to do additional research. Osaka is a huge metropolitan area (some include Kyoto and Kobe in it; I do) with a 17 million population, not as large as Tokyo but larger than the New York metro area. It turns out that there are 5 headquarters and/ or studios of TV networks (not stations but networks covering much of Japan) in the city of Osaka:
Television Osaka - 1-2-18, Otemae, Chūō-ku, Osaka, Osaka Prefecture
MBS Studio in USJ - Universal Studios Japan, 1-33, Sakurajima Nichome, Konohana-ku, Osaka-shi
Asai Broadcasting Corp. - 1-30, Fukushima Itchome, Fukushima-ku, Osaka-shi, (Hotarumachi)
Kansai Telecasting Corp. -1-7, Ogimachi Nichome, Kita-ku, Osaka
Yomiuri Telecasting Corp. - 2-33, Shiromi Nichome, Chūō-ku, Osaka-shi
I believe that the game show was filmed in one of the above.