With the tush push being so effective, they effectively only have to get 9 yards every three downs. So what kind of advantage does it give them
Part of it is modern defenses are not built to stop every play. They have a certain amount of bend to then to not give up bigger plays.
So, what you’ll often see from the Eagles, is they take those 3-5 yard buffer plays (a quick checkdown to Smith, a run up the middle). Then on third downs, defenses usually play the sticks, trying to force a punt. The Eagles push this to take the easy yards and come up short on 3rd down, specifically knowing how common 4th-1s can be.
What you saw in the Jets game (and early on against Kansas City and Buffalo to a degree), was the defense selling out to stop those little plays - extra pressure on first down, right coverage combined with blitzes. The goal was to set up 4th and long, rather than just 4th down.