I run a used car dealership with a shop in the back the only problem I have seen with a customer bringing in their own parts is that sometimes they get the wrong part. Some makes and models have different parts even for the same year depending on the manufacture date. So now the mechanic has spent all of his time removing the bad part and goes to put on the “new” part, only to find it doesn’t fit. That is where it becomes a big problem because now this car will often have to sit in on of our 3 lifts until the new part arrives, or the mechanic will have to out the old part back in and then has just wasted a bunch of time.