That's a good way to think about it. I couldn't think of a rule for when I drop the article. I guess for the state in the US we would say "hospitalized". So I would say "he was hospitalized", unless the identity of "they" in the original quote was important.
If you refer to the place by its name, then I think you can drop the article. e.g. "I went to Burger King"; saying "I went to the Burger King" would be fine on its own but a bit odd if you're not going to add further clarifying information suggested by your use of "the", such as "... on main street (as opposed to the one on 5th)". Similarly, "I went to Overlake Medical Center (the nearest hospital)".
"Burgle" is not a particularly common verb (746k results for "burgle" with quotes on Google) compared to "burglar" (40m), so it's not on the tip of the tongue.
The American instinct is then to reconstruct a verb from the more common noun, and the -ize pattern fits.
I'd guess that if you asked an American, they'd say that a burglar robs.
