Sometimes seemingly "irrelevant" songs are there for a theatrical purpose.
Little Lamb serves a small plot purpose showing Louise left alone and ignored on her birthday, but in the original it was a quiet respite for the audience after several noisy Ethel Merman numbers and the noisy vaudeville routines. Little Lamb gives the audience a chance to catch its breath and without it Act I would be too relentless.
CALL ME MADAM had a second act showstopper "They like Ike" that roused the audiences then because Eisenhower fever was just starting in the USA. (It was later used as his campaign song which must have pleased the staunchly republican Irving Berlin and Ethel Merman!) By 1953 it was out of date and easily dropped from the movie version.
In older musicals, potential showstoppers or hit songs were sometimes wedged into the plot often done with the excuse that it’s a song as part of an "entertainment" staged within the show.
Steam Heat, is presented as an entertainment at the union rally, gave Bob Fosse a chance to really show what he could do. And it does give a lift to the opening of Act II which tends to get bogged down in its plot until they go to Hernando's Hideaway. Ole!
I rather dislike shows that take the easy way out and use the songs as part of a show-within-a-show. 42ND STREET does that with almost all of its songs. In fact the number in 42ND STREET that has the most impact for me is "Lullaby of Broadway" because it comes from the characters. Also it's a great song.
Cast albums are NOT "soundtracks."
Live theatre does not use a "soundtrack." If it did, it wouldn't be live theatre!
I host a weekly one-hour radio program featuring cast album selections as well as songs by cabaret, jazz and theatre artists. The program, FRONT ROW CENTRE is heard Sundays 9 to 10 am and also Saturdays from 8 to 9 am (eastern times) on www.proudfm.com