How about this, any migrant caught on the US side of the border without the ability to show they came through a point of entry automatically loses their right to claim asylum and therefore can be immediately deported.

Again, you have to change the laws to do that. The executive can't just do that on their own.

Such a law isn't likely to pass Congress, either. You're thinking of random asylees today, but what drives the policy debate is a desire to avoid repeating the MS St. Louis incident - where a boatful of Jews seeking asylum from Nazi Germany was turned away. What differentiates asylum from regular immigration requests is that the asylee's home country is persecuting them, and might simply kill them if they are forced to go back. That's why asylees aren't deported until they get their hearing - because if they were at risk of being harmed in their home country, and you sent them back anyway until the hearing could happen, you've basically defeated the purpose of the hearing.

Wikipedia: MS St. Louis - Wikipedia