I do *not* recommend doing a pigtail from a Wall Connector. In addition to limiting you to 50a (40a usable) and being against the install instructions / code, it also bypasses a key safety feature (the temp sensor in the plug end). I recently helped someone else on the forum that did this and they nearly had a fire since the receptacle was wired wrong (insulation pinched in the terminal). Luckily he was monitoring his new install closely and detected excessive heat.
Note that even if you hard wire you can absolutely still take it with you when you leave... Just cap the wires and wall plate over them (or replace with a 14-50 or 6-50 receptacle with a 50a breaker).
You are right about needing 6 AWG wire for a 50a circuit if it is NM cable (Romex). 4 AWG would be required if you wanted to support a 60a breaker and still use Romex. Note that the Wall Connector has zero use for a neutral wire and it would just be in the way. They do make 4awg romex. 4-3 W/G NM-B Wire Black
I installed my Wall Connector with conduit which allows you to use the 75c insulation rating which gives you the ability to do 60a on 6 AWG (that is what I did). Though now that I think about it only has 5 amps of headroom to use on a 60a circuit (48a continuous draw by Model 3). So technically my install is only good to about 100 degrees Fahrenheit ambient temp.
THHN is what you want for use in conduit (usually dual rated as THWN or THWN-2).
The conduit could be EMT or plastic. Many people find the plastic easier to work with, but I vastly prefer EMT. I used EMT.
You could just use "flex" as well. Just make sure to use the proper fittings on the end with bushings to make sure the sharp Flex ends don't cut the wire.
To All-
Thank you very much for your instructive responses. They were all very helpful.