It’s a legendary variety show in which Downtown led a team of younger comedians on all sorts of challenges and projects.
You can watch some of the subbed episodes here but the show ran for a long time and most of the episodes are untranslated:
https://www.reddit.com/r/GakiNoTsukai/wiki/lincoln
You are probably misperceiving how popular the members of the Lincoln cast were at the time because of how popular they got after the show.
Downtown were the exception, they were definitely more popular than the other members at the time. The new cast has Kamaitachi, I would define them as skilled but they are nowhere close to how popular Downtown were at the time, what surprises me is that they didn’t chose Chidori as leaders, probably they are way too busy to start another program.
I forgot to add the name of the show in the title, it’s called Johnson (ジョンソン):
https://www.tbs.co.jp/johnson_tbs/
None, it’s from Gotsu Ee Kanji.