I've never worked for a Mgr (regional or property)that offered lunches outside of corporate parties. That's hitting my motivational sweet spot!
Based upon what you state, then the issue is identified. The Tech's have more personal investment at the other properties. (All things being equal, the described issue of the Dishwashers were definitely prioritized improperly. The the turn should been before them and after the H2O Heater.)
In simple terms I could see a couple of possible solutions (carrot and stick options). One method might be to structure, or restructure, an incentive program making it equally important to complete tasks on all properties, or weighted toward the property that needs the attention. That's the carrot.
The stick could be something like to remove a contractor performed task at the other properties and bring it in house there. Something like Maintenance installs flooring, performs carpet cleaning, painting, roofing, or other tasks that was assigned to contractors for the sake of time management. The resources spent on the contractors there, are now spent at the property that is having issues to ensure that tasks are completed on time. (There may be contract issues that may need to be addressed to make this happen if there are different owners involved.)
Over all it seems to me that you are asking the right questions. At some point it may have to move to a "soft stick" approach of: "This is the job. All 3 properties are our responsibility. We are going to prioritize like this.... Any questions?" (the hard stick approach is Do it or your fired... I'm not a fan)
I wish you well...