What! 
We’ve been conned.
It is possible to do BLFC that way, but I don’t think it’s worth the bother (having just done it as a test… 
). You still need to go to the Fridge entrance to turn in Rising Action and progress the quest marker. Then you can fast travel to Lynchwood, say Heyooo! to Steve as you drive into the Highlands, circle around, and get to the Gluttonous Thresher area. If old G-Thresh was there, that would be a neat alternate path to doing the quest, but it’s not:
You can just run the Highlands Outwash map backwards, which is interesting to do (once). Then turn around and come back, and the constructor and thresher will spawn. The Fridge is not my favorite map, but I don’t consider it worth this much effort to avoid. But there is definitely some novelty value to doing the quest in such a bizarre way. 
“Just sprint straight through the Fridge because that place is f****** horrible.”
I’ve gone through that exit, but never noticed that. That’s hilarious.
As a side note, seeing that little-used exit to the Highlands reminded me of a BL1 playthrough where I never used fast travel unless absolutely necessary (I can’t remember right now if I even absolutely needed to use it at any point); I drove everywhere. Definitely more time consuming but it made the game feel more road-trippy and it was kinda fun for (quite literally) a change of pace.
I might try that with BL2 sometime, though FT is indeed necessary to get to Sanctuary (obviously) after a certain point. I’ll make it a rule that I have to drive to the FT station at Three Horns nearest to where Sanctuary was on the ground to go to Sanctuary and back. 