We did not use uber to MK, but yes the Contemporary would have been the choice there.
Actually, the uber drivers picked us up at our building at SSR twice, and once from the main lobby of SSR (that's probably the easiest by explanation to the drivers). Also we used it from DS once when we exited from the marketplace and they picked us up there.
We used uber to get to the Grand Floidian once. And at AK and EP the uber drivers left us off at the entrance ( or certainly close to it). Just like if you would have used a taxi, I guess. We didn't go to HS.
I would say we never had to wait more then 5 minutes for a driver.
Like I said, if we were in a hurry to make a reservation somewhere, we took uber. Otherwise the Congress Park bus stop was near by and it was no big deal. I'm used to taking the buses at Disney.
You shouldn't worry about the transportation - delays can happen at any resort. But as far as being concerned about it prior to your trip, I really don't think it's necessary : )
I have a friend who is at SSR now and I'm so jealous. Have fun!