The 71 kWh added figure in the Remote S app is directly from Tesla's servers (and so is the rated/ideal miles added). I make no calculations or guesses on that. There is a way that this number could be wrong. And it's if the charge stopped and restarted. But what you stated is quite similar to what others have stated: The charge from 1% to 100% is not exactly 99% of their battery capacity. If I recall someone said he only added 81 kWh when he went from 1% to 100% on his 90D battery.
I have several guesses on the reasons on why this is:
1) 100% isn't always 100%. There were times when the charge said 100%, but I was able to get a few more rated miles in by simply waiting.
2) 1% isn't exactly 1%. I suspect that the Tesla has some "reserve miles" as some drivers reported driving at 0%. And also, the car probably shuts off the car before you fully drain the car to prevent damage to the batteries.
3) Tesla's servers or your car is reporting inaccurate data.
4) Battery degradation.
I think reason 2 is the major one, and here's the math on why I think so: When I charge to 100%, it says that I have 257 rated miles on my P85D. But as I'm charging, I noticed that the rated miles go up by 3.23 rated miles per kWh added. This 3.23 number is pretty consistent whether I add 50 miles or 100 miles. But if you divide 257 rated miles by 3.23, you only end up with 79.6 kWh. So a 100% charge seems to be missing 5.4 kWh. It's very odd. When I look at the press release for the P85D, it says that it has a range of 275 miles. If I divide 275 miles by 3.23, I get 85 kWh. So my conclusion is that the car is hiding a few kWh from our view and that 0% to 100% is more like 5% to 99% of battery capacity. I don't dare test out my theory, though, because I don't want to drive at 0% charge to find out how much the reserve actually is.