1. They didn't say ANY refugees in the country is entitled thinking. They meant as MANY as possible is entitled. Wich is a reasonable take.
2. Also, just because a UNHCR convention says it, doesn't mean it isn't entitled. (I would say the UN is an exemplar of entitlement. The US provides the largest budget while countries like China get an equal vote in vetoing.)
3. Everyone coming illegally is either a) a refugee, b) an economic migrant or c) a threat. Every country has the right to a reasonable process to determine which bucket someone falls into. Even the 1951 convention you linked allows for that. The US's process is actually less onereous than the EU's. The EU won't grant you asylum if you come through a safe country. The US has no such concept. You can be a single military aged male from China but you will still be validly considered for asylum just like a child or 80 year old from Afghanistan.
The entitled line of thinking is that everyone who comes from a country where things are shitty is a "refugee". That's the problem with the way this discussion is framed.
[1] https://www.unhcr.org/about-unhcr/who-we-are/1951-refugee-co...