And once the person (who is demonstrably able and willing to enter the country illegally) receives their 3k, is there anything to prevent them from just.....returning?

If the Democrats take control of Congress in about a year and Jeffries and Schumer haven't been primaried, then they almost certainly will be the leaders of both bodies, and based on all available evidence they will take that opportunity to immediately surrender in the name of unity and comity.