Okay, given that the guy who tried to repair the boiler was replacing the thermostat inside the tank (but that hadn't helped things) I figured I'd check to see if the connectors on it were corroded or anything. I sanded them down a bit and now the thermostat seems to be more sensible. i.e. it...
Hmm. It may have been called a pressure sensor then?
It may also not have been an electric element, but the boiler seemed able to keep water hot without the burner coming on.
Obviously I am far from an expert about boilers, but do have a good understanding of mechanics.
My boiler is an Ariston Genus storage gas combi (will find exact model when I get home tonight).
Just talking about it being set on DHW to begin with.
Originally the gas burner would not stay lit for more than 5-10 seconds, so the thing had been set on a really high temperature for it to...