If I might nudge your way of thinking, yes, that's kinda what I was thinking... but didn't actually say.
The tree very well could have just grown there as a volunteer, or planted by someone that didn't know what he was planting. Lots of people grow trees for showy, inedible fruits: an ornamental crabapple for example.
If you really want an invasive buckthorn, grow a male that won't spread seed. And yes, cut down the female buckthorn that does spread seed. If you live in Canada or the USA, the tree is probably listed on the state or province's prohibited species list. It's a tree from Europe and Asia, not North America.