To a American or Canadian ear, "burgled" sounds like something made-up through a bit of whimsical word-play.
The key difference with your example is that "robber", the noun, is derived from "rob", the verb. (Yes, it's an historical thing.)
With "burglar", the noun came first -- it's not derived from "burgle"; rather, "burgle" is a
back-formation based upon it.
But back-formations are much more the exception than the norm when it comes to deriving verbs from English nouns. The more customary way to derive a verb is to add -ize (or -ise) to the end. Thus, "burglarize".
What's interesting is how, when American and British English really started to diverge, "burglar" was a part of the lexicon but a verb form of the word was not. The route each took in doing so is perfectly legitimate -- neither inherently more "right" than the other -- but a century later, the alternate word from the other side of the pond seems silly, for both sides.
Of course, one can question the necessity of
any verb form, given lots of alternate words like rob, stole, broke into, etc. Off the top of my head, I'd say burglarize / burgle are more specific in their connotation, used normally in the passive voice, to refer to the specific act of person(s) entering a building (typically a domicile), in secret and without permission, and stealing items from within.
Ex:
"I was robbed" versus "I was burglarized". In the absence of any other context, the latter paints more detailed picture -- we can safely assume several more details about how and where the theft occurred from "burglarized" than we could from "robbed".