Yeah, I had the power button crack along with the raised screen on the right side with my launch switch. I sent it in for repair and they told me it would be about a week turnaround time. I got it back 8 days after I shipped it. But if they told you 2-3 weeks, I'm not sure then. My repair was in April, so maybe they were still doing quicker repairs for the launch period.
The problem though, was that they sent me back a unit that was warped and that had a raised screen on the bottom side. So I sent it back in for repair again, and got a unit that has a raised screen on the right side again.
So, at least in my personal experience, and as much as I love the Switch, the hardware issues have definitely bummed me out. I'm aware I could just be unlucky, but I wish United Radio would be bothered to do a more thorough check before sending out a unit with more problems. Since you're just getting the backplate replaced though, that should be a pretty safe repair I would hope. I doubt they'll send back a new system unless it had more issues like mine did.