That is where Boehner was a failure. IIRC, the main issue with all of the immigration reform was that enforcement was always to follow some sort of amnesty. You need to get that hammered out. The public is extremely amenable to tighter border control, you have to make the case to increase public pressure.
At a certain point in a right leaning coalition, you have to be as willing to fight for some of the right leaning principles as you are willing to capitulate to the opposition. Boehner had it harder because he had Obama at the top. We got nothing but tax reform out of the first two years of Trump.
Edit to add: And you still didn't answer the bigger question, what is the establishment GOP's plan for any of this?
Again, I go back to.... what can get 60 and 218? Unless you can get Senators to abolish the filibuster, those are the hurdles you must hit to do anything.
The Trump era got through a tax bill as it was something that could get through via reconciliation and a 50 vote hurdle. That's a once per year shot and any immigration related language wouldn't be permitted in such a bill.