Why can’t I just use a conventional three pin plug to charge my car?

This is an option but there are things to bare in mind when charging this way:

Your charging cable may not have the built in safety devices that would disconnect the car in the event of an electrical fault, generally a Mode 3 EV charger would have this either built in or part of the installation process.

It could possibly take twice as long to charge a fully electric car using this method as you would be limited to the 13amps supply from the socket outlet being used.

Although, a socket outlet is rated to 13 amps of current, they may not have been designed to be run at the limit for 10-20 hours constantly, and used 2-4 times per week.  This could cause overheating and even burning of the socket/plug top, which can be very risky.  A mode 3 EV charger has been designed for this type of use.