just to be clear, we don't have millions of illegal immigrants coming in each year, we have millions of encounters with illegal immigrants at the border. The number of illegal immigrants who actually make it past the southern border into the US is orders of magnitude smaller. Most illegal immigrants in the country today are people who came legally and are overstaying their visa. That's why we don't just give pro athletes visas all willy nilly.
The only 'illegal' immigrants who aren't turned away are those who claim they're seeking asylum. I put illegal in quotes because people seeking asylum are following a legal process, but the issue we're facing now is that people who are not actually seeking asylum are told they need to say they're seek asylum, and international law dictates that you have to process asylum seekers. This has caused our immigration courts to be backed up by multiple years, and the holding facilities where illegal immigrants would go to be processed are full, which is why the "catch and release" policy is in place. We literally don't have anywhere to put them.
The bipartisan bill that Trump told Republicans to back out on would have addressed these issues by capping the number of people who could enter, and by giving way more resources (border officers, asylum agents, judges, money) to the courts so they can work through the backlog we already have. It also would have effectively ended catch and release The cap was an average of 5000 encounters/day for a week, or 8500 in a single day. Again, it's not that 5000 people would be let in every day, it's that 5000 people max would be detained, processed and deported (or granted asylum if they qualified). The rest would just be detained and deported. It's my understanding that this would go against international law, but I mean, it's not like anyone would be able to stop us. This also wasn't a problem when the asylum process started.