One surface level comment. As a non-religious person, I have always found the distinction between buildings for different types of religions as being somewhat artificial. Sure, they have different architectural styles, but they all have a similar purpose in my mind.
So a mosque is a church is a temple, on some level.
Your house is the same as your friend's house, but you'd still rather have separate houses.
What you are saying is that any large open space could function for any sport. But that ignores how football teams need goal posts and a field that is a specific size and needs specific markings on the ground. And a soccer field is a different size and needs soccer goals. And basketball needs a hardwood floor and hoops that are certain distance apart and special markings on the ground. And bowling needs lanes and balls and pins. And tennis needs something else. And so on and so forth.
Religions are far far far more complex than any sport and no large open space could function any more than any large open space could function for all sports.