A refurb in this case likely covers remanufactured product as well. I can understand the concern seeing as companies don't really discuss their policies and programs around this, but having connections to a couple companies using refurbished/remanufactured product as a way to cut waste costs, it is usually better than you think.
Say I'm Apple and get your damaged iPhone. I would "salvage it" for useful parts (display if in good shape, logic board, etc), testing the parts as part of the process. Anything scratched, dinged, or non-functional goes into the recycle pile along with the battery. Now do this a hundred times, and/or get parts from the factory, and you have a supply of parts built up. Get some new batteries, and reassemble the unit from the parts bins you've built up and you've got a remanufactured unit.
Another way to go is have new parts shipped in from the factory, including batteries, and replace anything dinged/scratched/non-functional in the returned unit, along with the battery. Once the repair is done, box it and you've got a refurbished unit. This is similar to a repair, but with a little extra to make it "Like New" condition.
Now, I don't know which process Apple uses for the iPhone, but because the unit goes through testing either to be salvaged or repaired/refurbished, it's actually getting better QA coverage than with a unit off the manufacturing line. There they sample the batches and test some small random percentage of each batch.
Is the process perfect? No, some bad units may still get through for a variety of reasons including the tech simply having a bad day and not being at their best. However, the defect rate should be on par, or better, with new with QA being done on each part versus batch sampling. And the defect rate after refurb should be the same as the defect rate after repair. The process is the same, but the refurb contains a couple steps a repair doesn't to 'refresh' the unit and make it "Like New".
I've bough refurb from Apple for the last 20 years when I could. When I didn't, it was usually new models. I've had better luck with the refurbs than new. Although my sample size is only about a dozen or so, take my anecdote with a grain of salt. But I'd certainly be happy with a refurb replacement if I ever needed one. Doubly so if the SSD is easy to swap. Restoring a replacement iPhone takes forever for me when it comes to getting the music re-sync'd over.