The Dutch wiki suggests:
Afkomstig van het Nederduitse woord "brüden" (= plagen, lastig vallen, feitelijk: ontucht drijven met). [derived from the Low German word "brüden" (to plague, to bother, actually: to fornicate with)]
I can't confirm this exact verb but I suspect what they mean is the word I pronounce as "bröden". It is cognate to English "to breed" and "to brood" and can mean both. The Standard German cognate is "brüten" and has the same meaning as the Low German one.
In any case though, if that etymology is true, I would have thought that there should be an archaic form bryda, bryder. The past is indeed brydde (with an additional "d" which might come from an earlier bryd- stem. I can't find any sources to confirm it though, so take it with a grain of salt. Then again, even if it's ultimately false, it can still serve as a mnemonic aid I guess ;)
Because att bry sig doesn’t refer to the action of “caring about” as in “to take care of”. It’s the mental disposition “to care about” as in “to take strong interest in the well-being of”.
So if jag bryr mig om dig, I’m not saying that I’m making food for you, making sure you are safe etc. It just means that your well-being is important to me.
vård is a noun: hälsovård “health care”, sjukvård “medical care”, at vara i behov av vård “to be in need of care”. It’s not used as a verb.
bry just happens to be negative. There are such verbs, particularly when it comes to verbs that describe emotional reactions which can’t be controlled. Compare how in English you can “enjoy something”, but if there is no object you cannot just “enjoy”; instead you “enjoy yourself”.