Running my EV charger's extension cord across the grass to reach my driveway — how much of a fire/safety risk is this really?

I had an extension cord burn out on me while charging. Wasn't even a cheap one. Get the outlet moved, or better yet put a hardwired charger where you park.

You're not overthinking it. Running at 6 gauge over 10 feet, the wire itself isn't the problem -- it's barely breaking a sweat. The problem is every plug-to-receptacle joint in the chain, and you've added one. 14-50s already fail at a surprising rate under EV loads, even when they're installed to code, because they were really designed for a dryer or range that runs for an hour and shuts off, not 32-40A for six hours every single night. Now you have two of those connections instead of one, and you're plugging and unplugging one of them daily, which wears the contacts. If I were you I wouldn't "move the outlet" at all -- I'd skip the outlet. Have the electrician extend the circuit through the wall to the driveway side and hardwire an EVSE there. No receptacle, no GFCI breaker (which is required for the receptacle and isn't cheap), and most wall units come with an 18-25 ft cable, so where you park stops mattering much. In the meantime, at least turn the mobile charger down to 24A or so if your car lets you set it. Less current means less heat at the connections.