Is the United States supposed to take in an unlimited number of unauthorized border-crossers every day?

yes.

Is that the way it's going to go down? The right falsely accuses the left of being for "open borders", and the left's response is to actually just take the bait and make that their stance?

no borders is and has been the leftists stance.

In 1933, and again in 1937, the Roosevelt administration modified the “likely to become a public charge” restriction, but kept it in place. German Jews attempting to immigrate to the United States in the 1930s were still often rejected for economic reasons.

As a result of these legal and administrative obstacles to immigration, less than 20% of the German quota was filled during FDR’s first term. The Roosevelt administration focused on domestic problems, chiefly combatting the Great Depression. Though Americans certainly had information about the threat Nazism posed to German Jews, few could have imagined the persecution would escalate to mass murder. FDR did not take significant action to aid German Jews, either by ordering a diplomatic protest or by publicly supporting increased immigration.

damn, sure sounds like letting people in who need help could have saved an awful lot of folk