It sounds like you've been thorough with your troubleshooting. It's not the speaker, nor is it the input cables/wires, nor is it the source. All that's left is the speaker wire between the amp and front left speaker or the amp itself. My money is on the former, the wire might be pinched somewhere or loose at either side.
That doesn't sound like an overly complex system. If a shop can't figure out what's wrong after four attempts that's probably not a good shop. Usually we'd tell you to take it back before messing with it because otherwise they can say it's a result of something you did, but it sounds like you've already brought it to their attention. If you can access the wires at the speaker and the amp be sure they're secure. You could try swapping the left and right channel outputs at the amp. If there's still crackling then it's the wiring, if the crackling switches places it's the amp. If you can't do this easily yourself have them check it. You paid for a working system, you should have a working system.
As far as getting a refund from the shop, that's not my wheelhouse. I'd try working it out amicably before escalating it.