See
page 6 of the good practice guide it is not law, and it isn't even part of the regulations as such, but the guide to the regulations went OTT, not got it to hand, but seem to remember it listed minimum numbers which was higher than any electrician was likely to fit, so it is not in general followed.
Nothing should burn out, does not matter how many extension leads are plugged into each other, but if running on the limit a slightly lose terminal will over heat, as it expands and contracts with the heat it compresses the wires and terminal then releases them again allowing it to become looser and looser, until they finally burn out. But in theory the fuse should rupture before any damage, however the 13 amp plug has been modified over the years, with finger protectors so it can't dissipate the heat as well as the original, even using white instead of brown reduces the ability to get rid of the heat from the fuse.
However most 13A appliances only draws full 13A for a short time, so it does not have the time to warm up. As you put more and more appliances on the plug so the time it is used for increases as it has no time to cool, and once a plug over heats in a socket it damages the spring grip on the pins on the socket, so once a plug over heats, the socket also needs changing.
It is really down to the guy doing the electrical installation condition report (EICR) as to if he included it in his report, clearly if he has not listed it the land lord would be unaware, tenants I think now have to be given a copy of the EICR but law Scotland, England and Wales slightly different so not sure on Scottish rules.
Likely better to ask please, rather than trying to force. And at the moment it will be a little different.