The friend who played me the album remembered what I was going on about when he saw the question. The answer is Cooked on Phonics by Punky Brüster. The grumbly voice is doing the narrative in parts and the story is about the formation of a fake punk band from a metal one.
This is the live version of the song "Never Let Go", from the DVD Camel: In From the Cold (2014).
In this YouTube video from the DVD ("Fox Hill"), the musicians, clothes, instruments, set, and lighting all look the same as in the OP video.
According to the above link to Discogs, the track list is:
1 The Great Marsh
You can try Take Me Home Country Roads compilation CD album released in 2005, with John Denver's song opening featuring Johnny Cash, and others...
Also Country's Greatest Hits LP was released in 1985 has the song in the 13th track.
For an extensive research, you can try Discogs.
So far, I found no evidences on Discogs or Wikipedia or Spotify that the band Apocalyptica is related to the song Lux Aeterna composed by Clint Mansell.
I think it is some kind of Mandela effect that probably started when someone posted a non official trailer of the movie Apocalypto that features that music. Someone else posted another non official ...
I think the song you mean may be "Rhythm of the Rain" by The Cascades. The start of the song exactly matches your description of roaring thunder following by raindrops-like piano rhythm. It is a hit song from the 1960s, so it matches your timeline as well.
It's tough to prove a negative in this case, but in looking through LP scans on Discogs of all Kansas's early albums up to and including 1979's Monolith, I haven't found any mention whatsoever of "White Rabbit". Most of the 1970s LPs do have "Special Thanks" type sections though, so perhaps he was thanked under a different name? I'm assuming "White Rabbit" ...
The first thing I thought of when you were describing this was "The Champion" by Carmen.
This 1985 hit is epic rock with a Christian message, and a musical-like feel. It describes a battle between Jesus and Lucifer in a storytelling style.
I am very sorry - it turned out that this isn'ta music album and this isn't any music group, but just a music selection on the social network, with attached this photo for decoration.
"P von B" -- this is a monogram of the author (Paul Von Borax) of the photo. Good job, by the way. Here is the link to the photo album.
Thank you all for your interest in the ...
I'm pretty sure the album I'm looking for is Fantastic Playroom by New Young Pony Club (NYPC).
Ice Cream is the specific song I was looking for.
This album cover doesn't match though. Maybe there was a single?
It took 5 seconds to find the album on Google based on your description of the cover (type "sentimental hits album", hit enter, look at images...), however it is a white heart on and blue background. The album name is No.1 Sentimental Hits.