There are a few different issues here:
Service Capacity
Any time you make a large increase in electricity usage you need to do a Load Calculation. This takes into account many things, including:
- Size of the building
- Major fixed appliances such as water heater, EV charging, clothes dryer, oven, cooktop, etc.
- Required kitchen, bathroom and possibly other circuits
- HVAC - largest of Heating or Air Conditioning, since they don't run at the same time
It is a relatively complex thing, as it has a number of things that increase or decrease the calculation in possibly unexpected ways.
It is NOT:
- Add up breaker handles - way too high (except in the case of some very small subpanels with specific dedicated circuits)
- Electric bill average usage - way too low
- Electric bill peak usage - that gets closer, but needs to be considered over a long period of time and even then may not be close to the real number
- Guessing/estimating
Google: NEC Load Calculation to get details.
So the first step is to take your service capacity (typical is 200A, but sometimes more and often less, particularly in older houses - presumably you have 200A service based on the main breaker) and subtract the current Load Calculation and see what is left. Is that enough for 240V @ 50A? If so, great, you can do your 6-50. Is it enough for 240V @ 40A? 30A? 20A? Any of those can be made to work and provide enough power for most people to charge EVs most of the time.
If your Load Calculation shows you already oversubscribed then you have to deal with that ("heavy up" or load shedding) first. If your Load Calculation doesn't have quite enough then load shedding is likely the way to go unless you have other reasons for a "heavy up".
Panel Space
Your panel is full. Almost every breaker already has half-size (tandem/double-stuff) breakers. Most new circuits require AFCI (almost always best in the panel), GFCI (OK at point of use for 120V but 240V always in the panel) or both. AFCI and GFCI breakers are generally (there are some exceptions) available only in full-size breakers. So even if you can manage to squeeze in this one additional circuit, you really are running out of space and a subpanel may make a LOT of sense. A subpanel doesn't increase your service capacity but it does let you put in more breakers/circuits, including full-size breakers where needed.
As far as I am concerned, if the panel manufacturer offers a legitimate UL or ETL listed 50A quad (probably 50A inner pair, 15A or 20A outer pair) that's perfectly safe. But it won't have GFCI.
Receptacle vs. Hardwired
You are correct that a 6-50 is fine for EV charging as there is no need for the neutral of a 14-50. However, a receptacle for almost any use now requires GFCI. That does depend on NEC version (I think it started in 2020 and states vary quite a bit in their adoption time of each NEC version). In addition, some jurisdictions have exceptions that may apply. So you need to find out your specific local requirements.
There are three advantages of hardwired over receptacle for EV charging:
- No GFCI required. The EVSE and car already take care of that, so no GFCI is needed in the panel/breaker.
- Heavy-duty 50A receptacles are expensive. Due to the continuous (hours at a time) nature of EV charging, heavy-duty really is recommended.
- Many EVSE devices (e.g., Tesla Wall Charger) are simply not designed for plug/cord connection.
Plug/cord/receptacle makes sense if you have to move things. It doesn't make sense for EVSE that is installed once and left in place for years at a time.
If you say "I have a travel charger and want to plug that in", my answer is to keep that in the trunk or frunk for when you are on the road visiting someplace that doesn't have any hardwired EVSE.