Biden doesn't have a solution because the Executive Branch is not the problem. The fundamental problem is that are immigration laws are woefully outdated and completely incompatible with this current situation. The specific immediate problem is that the legal system put in place to lawfully manage this issue is so overburdened that a non eligible asylum seeker can stay in the country for years before their cases are adjudicated. That means that even if they are ultimately rejected, they can still work here to take money back home for months.

I actually do have a solution. The US should militarize the issue. Now, I'm not talking about troops or border walls. What we should do is draft all the lawyers into the Coast Guard and set up a special court system to process claims. Assign some as judges and advocates for each side and wind down the backlog. Of course this will never happen because wealthy and powerful people in our society will never be forced into service unlike the rest of us who they will happily push out in front of machine guns to die for our country.

Even if we don't go with the draft, the ultimate solution must come from Congress.