What size circuit do you have on your service to feed the outlet/connection? That's the half the key to your answer. Do you have a 100 amp circuit in the garage? The other half of the key, what max rate can the car charged to - 32, 40, 48, 72 or 80 amps when connected? Its the 80% rule. You can only draw 80% safely whatever the circuit is. So 50 amp circuit, 40 amp draw for charging. 40 amp circuit, 32 amp draw for charging. 100 amp circuit, 80 (or less) amp for charging. Can the car charge at 80 amps?
If you can't charge to the car faster than 40, then yes, 50 amp breaker is perfect for 50 amp circuit or even 100 amp circuit. But if the car can't charge faster than 40, then there's no point (not wise) to use higher than 40 amp.
If your Tesla can charge at 48 amp, and you want to maximize charging, what's the amperage circuit you need? 60 amp? Yes, 60 amp. The Mobile Connector with the car will work for that, so you don't "need" the wall connector. The Mobile Connector that comes with the car can charge whatever the car can handle.
If you need to charge some place else, like parents or in-laws or while traveling, and then you want to take your Mobile Connector and keep it in the car. (Its not a good idea to routinely/frequently disconnect/unplug your Mobile Connector from your NEMA 14-50. It wears out the adapter/contacts.) So, its wise to have another connector/cable for charging? So you are shopping for another connector/cable. Which is the better choice?
The Wall connector can do all. Let's say your home can only safely handle a 50 amp circuit for charging. Then there's no gain in getting a Wall Connector. Wall Connector and Mobile Connector are generally pretty close in price. Rule that out as a factor.
There's a remote disadvantage to installing the wall connector if you don't benefit from it. Remote chance that your installed (permanently hardwired) Wall Connector has a problem. You can't just unplug the Wall Connector and get your Mobile Connector out of your car to test if its connector/cable or car or outlet. If your primary charging cable is the Mobile Connector and you think it has a problem, can swap the home one with the one in the car, or you can take both and move it to another place to test it and the car.
I live in Hawaii. Cold or charging when its cold is not an issue for me. You live in Denver? Doesn't it get cold there? In cold weather, you don't want your car to start charging at 11:00 pm and finish at 3 am and the battery to start cooling down for your departure at 6:00 am. You want it the battery to warm up and charge longer - slower, so it finishing or just finished as close to your departure time, so the battery is still warm and does not need to use stored energy to warm up again. So, charging faster for shorter is not to your benefit or health of the car.
Early on some of us thought we could charge faster with dual chargers in the S. So we ordered dual chargers (paid extra). Then we learned we needed a 100 amp circuit to get the benefit of dual chargers to shave 1-2 hours or part off the charging time. But if you plug the car in at night at 11:00 pm and it finishes at 3 am or 4:30 am what's it matter 95-98% of the time if you don't leave the house until 6:00 am. Then we checked the cost of adding to our service to be able to install a more costly installed 100 amp circuit - thousands. Thousands of additional cost to upgrade house and circuity to shave an hour or two once in a while? Service or Circuit upgrade is not cost effective.
And the car still only charged at 40 amps with dual chargers when the circuit was 50 amp. Dual chargers were a waste of money for many of us. And back then, the Wall Connector was $1500, the Mobile Connector was $650, (I think).
Questions?