It’s not nearly as contagious as you’re making it out to be. I’m a nurse who directly cares for patients with MRSA. Nosocomal infections are a major issue and require contact isolation to prevent but MRSA in general is not a particularly scary pathogen. If you wanted to deliberately infect someone, you’d have to straight up inject them with it.
Staph aureus is already everywhere and even community aquired MRSA is becoming common.
There was a time when we screened everybody, but now we don’t even do that. It ended up being considered a waste of contact isolation gear and carer time to gown up before you entered 1 in 5 rooms that just happened to have a positive MRSA skin swab.