no, the hospital analogy is almost perfect.

suppose somebody found a way in through the back door of a hospital and snuck into an empty room. when discovered, what would happen?

the doctor would no doubt make a snap triage decision; if they had a knife in their neck or something, then, yeah, they'd need to be dealt with first. otherwise, they're going to receive a warning not to do that again, and be escorted back to the waiting room - meaning they're going to have to wait even longer than they would have if they just did it right in the first place.

all i'm really saying is that we should be properly triaging refugees, which means bringing them back to the waiting room when they sneak into the country via the back door.

.....and, that is a fundamentally egalitarian and inherently left-wing position.