>> How would having a trademark change anything?
It would change a LOT. For example, if the trademark was already in place prior to the current .org owner registering or purchasing the domain (perhaps when it expired), then you could show that the registration of the .org or the purchase of the .org domain was done "in bad faith", and a UDRP could be filed. There would be the likelihood of them winning the case if they could show "bad faith" registration or purchasing of the domain.
Proving "bad faith" in registration could potentially be done without a trademark in place, but it does help.
The content of the website doesn't necessarily matter at all.