Why use Tesla wall connector in a first place and not just the standard nema/dryer outlet with the Tesla cord/charger? It seems like people are overpaying for nothing.
Breaker size =/= charging speed. Breakers are oversized for safety reasons. The Wall Connector is on a 60A breaker and charges at 48A. NEMA 14-50 outlets are on 50A breakers but can't charge your car at 48A. 40 iirc, and the mobile connector that comes with the car maxes out at 32A.
I've had both setups and whether the full amperage charger is worth it or not depends on your use case. If you're just going to plug it in overnight, it doesn't matter. It's about an hour's difference for a top up. If you're going to wait for your car to charge and unplug it because you share the charger or have to run an extension over a public sidewalk, then the faster speed is worth it.
There are a lot of different mobile chargers, if you don't like the specs on the Tesla charger buy a different one. Though do beware that cheap 50A receptacles cannot handle 50 amps continuous. They are for stoves (max 40A), or welders (low duty cycle since you spend more time in setup then welding - assembly lines use better receptacles)
I can confirm this. Our Model 3 doesn't charge as fast using a NEMA 14-50 plug connected via the Tesla-provided mobile charger.
When we moved to a new house, we bought a Tesla wall charger, and it indeed charges at higher amps, but I don't know if the extra speed has necessarily been worth it since we primarily charge the car overnight.
You don't want to use a standard outlet, since it's not designed to handle full current(?) for hours. There are special outlets for EV charging, and they work outdoors. Just be very sure to have a GFCI breaker behind it.