They cease to be refugees when they have settled somewhere or voluntarily returned to their country of origin. See the 1951 UN Refugee Convention and the 1967 Protocol Relating to the Status of Refugees.
If they all settled at the point closest to their homeland that wasn’t “at war” I’m pretty sure there would soon be conflict there.
Britain is a rich, developed nation that is quite easily able to support our “fair share” of refugees along with the rest of Europe.
Combined with the argument you're making here it is just a cynical attempt to hijack people's good natured desire to help genuine war refugees, into jackhammering unlimited migration and open borders into society without consideration of any costs. This must not be allowed to happen as this kind of abuse will ultimately result in the collapse of support for any kind of refugee and asylum support.
However, it is a blank cheque once you’ve surmounted those challenges and have arrived at the safe country you have chosen (and were able to reach), though most countries receiving such refugees are doing their best not to honour this obligation.
It’s worth noting that the UK tries hard to prevent refugees reaching our shores, but the vast majority that do, and apply for asylum, are granted it [1].
https://www.gov.uk/government/statistics/immigration-statist...
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...