Alamo has a free shuttle that will pick you up anywhere on Disney property not just the TTC that the other poster has mentioned. YOu must call the CCC the day before to arrange a pick up location and time
Yes, they absolutely pick up at Pop, or anywhere onsite accessible to general traffic (so, TTC instead of MK, and FW Outpost instead of inside the resort.
There are also Alamo/National desks at the Walt Disney World Dolphin and the Hilton Buena Vista Palace. More often than not in my experience, the price is the same because the same people run all three locations. Depending on where you'll be on the day you pick up the car, one of those might be better than dealing with the shuttle to the Car Care Center.
For example, if you're going to be in World Showcase, you can simply walk to the Walt Disney World Dolphin. If you are going to be at Disney Springs (particularly the Marketplace section), the walk to the Hilton Buena Vista Palace isn't too bad.
They will pick you up at your hotel (or anywhere on disney property) and drop you off at your hotel (or anywhere on Disney property). I have done this a few times when I needed to get bloodwork done in the middle of a trip and now we do a one-way rental from the airport to our hotel. Cheaper than a town car and we stop at target on our way.
Also they will give you a pick-up time at your resort to get you to the CCC, you can't choose a time. When dropping off the car they take you back as soon as possible. I think I waited less than 5 min. When I dropped the car I was asked if I needed a ride back and the guy checking me out told someone to get the van to drive me back. Next thing I knew it was right there and off we went.
As an FYI for anybody that has to drop the car off after hours at the CCC and use a Lyft/Uber there is a Gas Station (Speedway?) there so not only can you fill up it's well lit and there are always people coming and going.