When they begin the beguine...
In American English, it is familiar that a sentence is started with "For". For example:
(1) For all the problems, it was a valuable experience.
(2) For one thing, we can't afford it.
These could both be dependent clauses following a comma and a previous dependent or independent clause. They don't stand alone very well.
In addition, your correction has what my teachers would have called awkward construction. "In American English, it is familiar....." appears or sounds more like the English of India, Americans would say "It's common usage..." and you don't need to capitalize "For", you need a comma.
"According to the AHD, starting a sentence with "for" is common usage, for example: ..."
I'd be interested to know where you learned English. There's a link in my Favorites to a guy doing research on dialect and usage. I'll add it here when I find it. In my neck of the woods, we bobble when we make errors and our mind boggles at our stupidity.
Something that is mind boggling is overwhelming but not necessarily in error.