The splitting of kettle and lead into two separate PAT testing items was what allowed us to get the numbers of tests per day required. One test but two labels. However they were given their own ID with the whole idea the lead and kettle could be separated and swapped around, the C15/16 and C15A/16A were the original kettle plug and sockets after the old round one, but latter the C13/14 started to be used. This caused a problem as by then traditionally these were used for computers typically with a 3 amp fuse.
Many of the leads used just 0.5 mm² cable and so it was important to ensure these only had 3 amp fuses fitted. In hind sight it was wrong to have a 10 amp socket on a cable unable to take that power, but in the beginning they could not be used with kettles so suppose no one thought about it. In the auto trade I noted again and again where a wire burnt out in a harness often in the harness it was OK only as it left the harness so no other cables to help remove the heat, did the cable burn, I would suspect the same with a moulded plug or socket, they can remove heat better than a rewire-able one, so there is some sense in saying moulded plugs can use thinner cable. However there is a difference between moulded and non re-useable the latter does not always have any material to help remove heat.
The other point is design current and fault current are also two different things, so with a 10A socket the design current is 10A even if the kettle takes 13A the design current is still 10A for the lead. A 13A fuse will rupture with a short circuit so will protect the cable, and however much we know kettles use more than 10A it is the kettle at fault not the cable. And for years there has been two preferred fuse sizes 3 and 13 amp so personally I see no real problem in fitting a 13 amp fuse into the plug with the cable shown.
If one was to be pedantic enough to fail the lead because it had a 13A instead of 10A fuse, you would have to also fail most kettles for drawing over 10A, some common sense must be used. I have just bought a new toy, a smart plug which measures power used, I also have a new kettle which meters out one cup of boiling water at a time, idea is one reduced time to boil and two less energy wasted, however I found that the unit uses over 13A common to see it show 3020 watt and the smart plug sensing overload would auto switch it off, tried other meters and yes it does use over 3kW, since only one cup of boiling water this slight overload is OK, and if not using a smart plug which trips out over 3 kW I would not have noticed. Rated at 3 kW but guess volts slightly over 230 hence overload. Even 3 kW is over 13A just at 230 volt.
I think some times we follow the rule book too closely, we are professionals and as such we should be able to judge if safe or not.