This year, I moved from the machine shop to quality. After 25 years on the floor, I was tired of the competition, and listening to all the machinists go on about how much better they were than the other machinists. We are an aerospace and defense supplier, and I doubled my pay by moving to quality. All of our quality engineers and inspectors are former machinists.
I've worked at a few different shops over the years, and I really think it has a lot to do with how your shop views quality. A machinist typically checks 10 or 12 critical locations at the most on a part and assumes the rest is good. In quality, we not only check those 10 or 12 locations, plus another 50, plus run it on the CMM and check another 10,000 points, check the hardness, check the wall thickness with ultrasound, check the surface finish with some fancy gadgets, check some locations on the optical comparator, and then record everything in extreme detail to fulfill our ISO standards. We can tell if an insert needs to be changed way before the machinist will ever be able to notice it, since our CMMs measure down to the 0.0000001", so we get with the machinist and get the problem fixed before it ever becomes an issue. Quality also has to see the bigger picture and account for what will happen to the part when it goes to the departments after the machine shop. A machinist might be running a part that measures good, but the humidity is affecting the paint shop and parts are borderline good after paint, or EDM is having some issues, then quality has the machinists make adjustments to minimize the issues.
On top of all that, we calibrate gauges on a regular basis. If parts are sent out for finishing, we measure them when they come back in. If we get parts in from a 3rd party supplier, then we measure all of those parts before they go to assembly.
Our machinists do program, check parts, record measurements, and all the other things that typical machinists do, and they do take responsibility for their parts, but quality ultimately has the final responsibility. If a part that isn't good makes it the DOD, then the operator will never hear about, but the quality inspector that signed off on it, will hear about it from every member of management.
If your inspectors just mic a few parts and do some paperwork, then your shop probably doesn't require the level of quality that ours does. I've worked at a shop where there was no quality, the machinist measured the parts and the next guy just assumed that they were good. Sometimes they were, and sometimes the machinist was just trying to get some parts out the door and get his paycheck.
At my current job, I guess we do "inspect quality into parts", because the machinists don't have the ability to measure to 0.0000001", they don't know how to program or set up a CMM, or a hardness gauge, or all the other gadgets in the quality lab. The machinists do what they do best, set up and run their machine and run what they believe is the best possible part that they can make. They get paid extremely well, based on what I've seen other shops paying. Quality just happens to make quite a bit more, because they take a lot more responsibility.
It depends on the shop. For some shops, the measurements from a caliper and mic are good enough, for other shops, it's a lot more precise.