I had exactly the same situation. Even a bit more extreme.
My barn, like yours, extended over 2 properties. At some point, 30 years ago, long before I bought it, the then neighbor simply cut through the roof battens and let his part fall into disrepair. Over the years, the roof and walls on my side then collapsed.
The partial demolition was carried out without securing my part of the barn. When we bought the property, a reconstruction or demolition was up for debate. The problem was, of course, that one could never again build such a structure. Border construction, wall height, etc. For that reason, we rebuilt the barn. In retrospect, a very good decision.
Since we considered it a repair service and actually restored the barn to its original state, we were reported to the building authority by the neighbors. 10 years after completion. Afterwards, we filed a building application and still have to install a fire protection wall or clad the gable with it. The demolition, which actually caused the collapse of our barn, remained without consequences.
Of course, my barn is a very beautiful old and solidly built building with a hipped roof and vaulted cellar as well as 2 gates wide enough for trucks to drive through.
Advantages?
Garage, material storage, workshop, wood storage, etc. I do everything from tire changing, wood cutting, garden parties, etc. in the barn. Completely weather-independent. There is enough roof area facing south for a photovoltaic system. The vaulted cellar has a constant 7 degrees for storing all kinds of food. The perfect storage temperature for beer.
Nice and cool in summer, protected in winter. Basically my second apartment.