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By checking the switched live to the boiler you are checking the thermostat.
The display can work fine but still not allow a current to pass through the unit when its supposed to.
I am not sure its possible to test these unless you can rig up a unit so that you can get a meter into the switched live prong while activating the stat.
My advice would be to buy a new one as im pretty sure its broken. If you buy it from a diy store you can always take it back if im wrong.