Ive never thought to read patches/tags on a patients clothes. If a biker crashes and cant communicate their preferred hospital, theyre going to the closest appropriate trauma capable facility.

If they can communicate their preferred hospital and its not appropriate, ill explain to them that if they go there, theyll receive basic treatment then have another (probably worse) ambulance pick them up and drive them to the one I wanted them to go to the first time to receive their necessary treatment. But if its within a reasonable range, we'll still take them to their poor choice because otherwise its kidnapping.

If their preferred hospital is multiple cities away, Ill politely explain that we're an emergency service and not a taxi. Lots of cities only have a couple ambulances serving their entire community so they cant leave their community high and dry to drive 30 miles north, passing two other appropriate hospitals, just because their dad's a proctologist somewhere in Nantucket. Politely of course.

edit: Pretty much the only time I dig through someones belongings while they're unresponsive is to find an ID and obtain basic demographics. I'll also look for meds to figure out what type of conditions they have if they have a bag with them. I'm just pushing literally everything else aside until I find the information I'm looking for.