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The rotor and caliper are definitely plausible but I'm not sure about the master cylinder.
The caliper is what holds the brake pads and squeezes them against the rotor when you press the brakes. It's not uncommon for a caliper to stick and hold pressure on the pads after you let off the brake and that will cause what you are having. This will damage the pads and rotors if it is driven much that way, so having to replace the rotor along with a dragging caliper isn't out of the question at all.
I'm not completely sold on it needing a master cylinder, at least without knowing why they want to replace it. If the master cylinder was causing the brakes to drag it wouldn't just be on one wheel, it would be a minimum of two and usually all four. I would try and find out why they want to replace the master cylinder before committing to it. There may be a completely unrelated issue, they may have found that it's leaking or something like that and aren't wanting to replace it because of the brake drag.