Most have few resources. The ones with more resources have more options about where they flee to.
I don't really have a dog in this fight one way or the other, but people like you need to realize this - that sort of attitude and activism will sooner or later mean the end of the asylum system for everyone. We are already seeing credible candidates for Prime Minister want to withdraw from the ECHR to make deportations for failed asylum possible (not even easy, just possible) and it's because the public has no patience for the endless flood of "refugees" who inexplicably find France unacceptable. Abuse of the system will lead to it being terminated, and that will be popular with voters.
I don't personally have a strong belief that refugees should have a blank cheque to settle anywhere. It might be fairer to permit refugees to travel further, as it's unclear why neighbouring countries should shoulder the burden. However, the only point I was making is that my understanding is that today they _do_ have that right, which is why countries make it so hard to arrive.
[1] https://www.gov.uk/government/statistics/immigration-statist...