Does that treaty oblige us to process as many as show up?

Yes, for people who present a prima facie claim that they meet the criteria. Just like the constitution obliges us to provide a trial to as many people commit crimes, the 1951 Convention guarantees refugees at least a fair hearing to make their case that they are genuinely and legitimately fleeing persecution that meets the international standard.

It's annoying and cumbersome to provide basic due process to people - to give a fair hearing to their claims. But it's a bare minimum obligation of basic humanity and decency, that someone is given at least one chance to make an argument on their behalf to someone with authority. This isn't Judge Dredd - even people that have obviously and irrefutably broken the law aren't dispensed summary justice on the street, but are entitled to have a process by which their guilt is determined. The same is true of one's status as an asylee (or any other claim that they should not be immediately deported). Just like the beat cop doesn't get to make the final decision on a suspect's guilt right there on the sidewalk, the Border Patrol agent doesn't get to make the final decision on someone's status right there on the bank of the river. They get to make their argument at a hearing.