My solution to the Tush Push is to have your 4 biggest/strongest Dee lineman in the A and B gaps, so they can shoot through the gaps at a low level for penetration, to neutralize Kelce.
And then take 2 more linemen (not LBs), and place them at the LB position up close in the A gaps. Then at the snap, they jump the pile and grab the QB, and try to bend him backward. Sure, the risk of QB injury will be high, with 600 or so pounds coming over the top. And all it will take is one key injury to Kelce or Hurtz because of that play, then they will choose something else. Of course, everywhere else, everyone is in man coverage, so they have to hold up on the off chance of a play action pass.
The key is you need Kelce neutralized, and 2 big boys jumping through the air to grab Hurtz.
To get the right personnel on the field, the defense may need to call a timeout. But for the entire Eagles game, I'd have these extra linemen standing near the field on the sideline with their helmets on ready to go in...because as soon as it's 4th and short (at a moment's notice), they will need to sprint onto the field to replace some combination of LBers and/or a safety. No time to sprint in from the bench area, they need to be standing and paying attention to the game...especially in the RZ.