@Mara Witt you have a lot of good advice and a bunch of different opinions regarding your current issues with your tenant. I want to comment on the heat issue, I have worked in the HVAC field for 15 years and I have a few questions and some pointers that might help you. First just a little clarification as to the system that is being used to heat this house, you referred to a "furnace" and "radiators" are these hot water radiators? if so the system used to heat the house is a boiler not a furnace, not a major point, I am just trying to get all the info on the system. As far as your handyman is concerned, the fact that he is not working for a HVAC company currently might be the reason it is taking him so long to get parts, I know not all parts are readily available all the time, but if its a heating unit of any kind there should be some place locally to get the parts in a day or two.
As far as his license is concerned, not all area require a license for heating, so people might be misunderstanding this, a license is required to work with the refrigerant gases used in Air Conditioning systems.
Is this heating system controlled by 1 thermostat? if so and it is a boiler system(I am making that assumption because you talk about radiators) and you have 2 radiators out of 6 that are not working it is very possible for some rooms to be getting very cold on cold nights. The system may be working fine but with a third of your heating output not working you could have cold spots in the house.
Here is something to consider if you are running on 1 thermostat you can consider installing a new wifi thermostat and you would have the ability to set up monitoring so when she tells you the house was "20" you can look back and tell her that it never went below 65 or whatever temp.