It's Dos-Mukasan in Latin [alphabet]. Unfortunately, most other stuff that I've heard from them isn't as good (unless you like that kind of [folk-]pop[-rock]).
About Czech music, well, what you've linked is avant-garde and sort of by definition outside mainstream. (Bittová is a sort of popular actress in a way, though.)
I guess one unusual thing was that plenty of musicians, who became successful state-approved pop singers during normalization
, started with "bigbeat" bands, often playing progressive/art rock or covers of classic Western rock with (often very well) rewritten localized lyrics (which they often tried even later in their official production)—a couple of examples:
Marie Rottrová: Lásko voníš deštěm (Black Sabbath: She's Gone)
Jiří Schelinger: Šípková růženka (Deep Purple: Soldier of Fortune)
Meanwhile, others stayed more or less underground and typically got banned at some point.
Flamengo: Kuře v hodinkách
Blue Effect: Čajovna
Synkopy 61: Xantipa