I am now convinced that you are right.
If your diocese flies the gay flag:
http://www.dioceseofcleveland.org/gayandlesbianfamilyministry/images/rnbwtile.gif
I assume thre is no problem if a Catholic family attend a traditional Chapel.
Exactly. The Vatican said that you can attend an SSPX chapel as long as you are not doing it with an intention to demonstrate adherence to schism. Far from wanting to adhere to schism, you are simply trying to protect your family from a diocese run by evil perverts.
I attended St. Peregrine's a half dozen times this summer, and it was wonderful. The priest is very holy, and the families are nice and friendly. It is a very reverent atmosphere, although the chapel is extremely small. They are in the process of building a larger place out in Brecksville. They also have a school.
So why try to re-invent the wheel, which is what these indult places are doing, when you already have the traditional Catholic faith and even a traditional Catholic school right in your own backyard?