- It is stated that since Ramsey and his crew are not members of Starfleet, they can't be commanded to leave Angel One. But Angel One is an inhabited planet and the inhabitants - at least the rulers - want them gone. Wouldn't that make them illegal immigrants? Why can't Angel One simply deport them and the Enterprise then transport them back to Federation territory? The way the story is written, it seems to be implied that Federation citizens has the right to settle anywhere, even on planets outside the Federation, regardless of whether or not the inhabitants want them there.
- The Angel One's government's position is that the Enterprise can take the men if they want to go, but if the men insist on staying, which they do, then they are to be executed. The problem here is that official extradition by the Angel One authorities is just not on the table if they conduct the arrest and the men refuse to volunteer to leave, and Angel One has not delegated the authority to arrest to the Enterprise crew. So without that official offer to arrest and extradite, the Enterprise crew's hands are tied. It is an exceptional case and a clear, albeit narrow, legal gap that the situation has fallen into. We have to remember that Starfleet policy is that local legal systems take precedence, and the local legal system here is taking a weird yet definite stance on the matter.
- Since there is no evidence in the episode that Angel One possesses spaceflight capability, it may be that their legal system is not designed to address situations involving non-natives. In Real Life, no nation on Earth has an extradition treaty with any alien civilization either. So the government of Angel One is operating in an ambiguous legal situation, which is most certainly why the head of state is personally involved rather than a judicial system. It is highly unlikely that Beata hears every criminal case on the planet. The men from the Odin are not legal citizens of Angel One, they are citizens of the Federation. But the Federation is being wishy-washy about enforcing the conduct of their citizens on a sovereign world that is not subject to Federation laws. The bigger question that this raises is how the Federation is able to enforce the Prime Directive at all. If it does not apply to civilians, only to Starfleet personnel, then anybody with a warp-capable shuttlecraft can potentially wreak havoc with any pre-warp society they can reach so long as they are not affiliated with Starfleet or the Federation government!
Headscratchers / Star Trek: The Next Generation S1E13 "Angel One"