I think the LR Model 3 can charge at up to 48 amps with the wall charger, but what you need to do is dependent on your driving profile. How many miles of range do you need to replenish, and over what time frame? My own driving was some 60 miles of range on an average day, and 24 amps has been plenty to replenish it. In fact, for the first two years I owned the car, I was sharing the clothes dryer's 30 amp outlet between the car (24 amp charging) and the laundry, and never had a problem getting a full charge whenever I needed it.
When I installed my 14-50 outlet last year, I upped the wire size just as you are suggesting, from the required #6 to #4 instead. That gives me an upgrade possibility, but it also makes the charging path just that much stiffer (less voltage drop). The run is some 30 ' between the panel and the plug; not a long distance, but it was a bit of peace of mind to have it. #4 was the largest wire size that the 14-50 outlet would accept, by the way, so there are limits to doing this. And the thicker wire was a royal pain to deal with.
So, bottom line, if putting in a higher capacity is not a big deal cost-wise, go for it. You can always select something less. But if it's going to require a complete panel upgrade, I'd back off a bit. A full 70 amps is seldom necessary at home.
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