Really? Intuitively it looks like it should be pronounced "Switch" to me.

Vowels in Welsh are not pronounced like they are in English.

(“Cwtch” is Welsh for “hug”. I know this because there's a good beer called Cwtch.)

One thing the Welsh language is not, is intuitive.

I don't know how either "butch" or "cutch" is pronounced. You might want to offer a common word for people who did not grow up in America...

edit: butcher?

Yes, butch is like butcher but without the "er"

This may help, although I would have thought butch was common enough. E.g. Butch Cassidy and the Sundance Kid

https://www.google.com/search?q=define+butch&oq=define+butch...

Like "clutch" without the "L"?

Close. This is from the homepage:

How do I pronounce Cwtch? Like "kutch", to rhyme with "butch".

In common use you might say "Cwtch in" to mean "snuggle in" or "cuddle in close'

That is a really damn cute name