That Follies so-called "overture" occurs after a lengthy prelude and even a shortish book scene, so it's not an overture in any real sense. The overture is by definition the first thing heard. Otherwise it isn't an overture.
It is called one in the score, because it's a medley and there really isn't any term for what it is except "Potpourri" or "Arrival of Guests," etc. It uses cut numbers, but many musicals have included cut numbers in their incidental music. You can hear the start of "Say a Prayer For Me Tonight" at the start of My Fair Lady's Embassy Ball sequence, for instance.
My Fair Lady's overture successfully captures the feel of the show, as does the "entr'acte". They're both tuneful and fun. I love them.
I'm partial to Camelot's majestic overture, and the Carousel waltz is brilliant.
For an exciting beginning, I don't think anything can compare to the original (NOT the current tour's) end of The Phantom of the Opera's prologue, followed by a loud bang, and the chandelier wiggling about and starting to rise as the music plays. I'm not sure what the music is called when it comes after a prologue -- I've heard it's not technically an overture. BTW, POTO's entr'acte is pretty darned impressive, too.
Audrey, the Phantom Phanatic, who nonetheless would rather be Jean Valjean, who knew how to make lemonade out of lemons.