Given the state of that 3 pin plug, then I would strongly suggest that you consider replacing the socket outlet front that it was plugged into.

good point.
Totally agree on the issue of BS1363 not being designed for sustained usage at high current. And that problem has got worse in the last 5 years or so when most EV's have at least 50kWh batteries compared to the "first generation" ones that were often in the 20-30kWh range, resulting in extended charging durations.
With regard to your other points... my 2p worth is:-
Surprised it never tripped the breaker tbh ?.
An MCB will only trip when significantly too much current is drawn and nothing to do with the lack of dissipation of heat on any part of a circuit. It was likely just 8 or 10A passing though the plug, with a poor fuse connection that has caused the overheat issue. I doubt the granny charger started pulling 40A+ that would be needed to trip a 32A ring final MCB.
Assuming your consumer board ( fuse box ) has them installed of course.
If not, then consider replacing the front with a socket accessory that does includes a built in RCD trip at the very least.
All CU's should have old fashioned fuses, if not MCBs.
With regard to RCD's; they were
mandatory for all new builds after mid 1970, (edited out) commonly installed for any CU upgrade since the 16th Edition Wiring regs. (=1992). RCDs then became mandatory for any new ring final upgrades after 2008.
So, it's technically possible the house was pre-1970's and never had update that added RCD protection to the ring circuits. If that
is the case, I'd suggest the OP gets a sparky into upgrade the electrics to current standards, rather than having a workaround of an individual RCD protected socket.