Tonsillitis caused by uncapped tooth?

No, It is Very Unlikely that it has something to do, normally a prepare tooth is kind of clean, never completely free of bacteria (as you know it habitats the mouth) but normally not to cause this type of infection.

It is unlikely that an uncapped tooth would cause tonsillitis. I would not think the two are linked at all.

Very unlikely that that could cause your issue. Most likely just a coincendence. Unless there was an infection under the crown to begin with.