The story doesn't explain how a spy working as a waiter, purchase manager or "artist" can gain access to military secrets.
Moreover, this was about establishing a long term cover. He progressed through his cover career until he became an art dealer where he most definitely would have been hob nobbing with powerful people. Even if he didn't have access to "secrets" per se, he would have had access to useful information that would otherwise be hard to obtain.
They didn't actually placed tapping devices in the Queen's furniture. From the article:
One of his first ideas was to try planting listening devices in the Queen's furniture, he recalls, though he and his bosses realised it was technically unrealistic.
(At least that's what I thought when reading it anyway.)
A hotel waiter or a concierge is in the perfect position to mingle with the rich and powerful.
Upper class people use hotels for extra-marital affairs and business trips in unknown cities. You do not need a first degree connection with the military to get an in.
The bar of a famous Dutch hotel was known to be frequented by American spies and drug agents looking to ensnare peons of the local drug maffia, working their way up from there.