Hi Jacob, welcome to permies!
Going by your diagram, your system will thermosiphon heat up to your radiators... I haven't the math to comment on how effectively it will do so, especially since more quite a bit more info (distances, tubing sizes, temperature deltas, etc) would be needed.
The risk of overheating is a real one depending on where exactly you mean that collector coil to go, and with less water you'd be able to boil it that much more quickly. A pressure relief valve seems kind of important, but a low-tech alternative would be a system that is not sealed, and is inherently unpressurized...
Would this collector system tie in to the main heating system for the house? Or would you be removing these radiators from the central heat system to dedicate them to this system? If so, would this be switched with valves, or a permanent reassignment?
What sort of corrosion prevention measures are used in your existing radiator based system? How will you address this in the new system? This could impact the effectiveness of a thermosiphon by increasing viscosity; also, I wouldn't want antifreeze-laden water evaporating into my living space, in the case of an unsealed system.
One challenge of building a system like your diagram is the difficulty of experimentation. You don't know how the final system will behave until the bench is built... then, if you find it is getting too much or too little heat, what do you do? Dismantle the bench to adjust? No thanks! I'd suggest you plan on prototyping for a while with some sort of placeholder bench that allowed for easy access to, and adjustments of, the internal copper coil. Perhaps insulate around the chimney a little, but then use removable mass; big bags of something that you can lift in/out of a temporary wooden-framed bench?
An alternative that I personally would probably prefer would be to build the
RMH, finish the bench, etc... then take on this
project as an add-on, mounting the heat collecting coil around/near the barrel, rather than inside the mass. This would only work when the RMH was burning... but I am a bit skeptical that '2 little radiators' will make much of a temperature difference upstairs when the water inside is only being heated by the warmish mass, rather than the hot barrel. Of
course it would be particularly important to consider the pressure issue here.
Another thing to consider is slight changes to airflow that may make a big difference to house comfort. Perhaps it is possible to add ventilation in the ceiling of the room with the RMH, to allow some of the hot air passage into the upper rooms directly?
Edit: and in the time it took me to say, "should do something, but I don't know how much", along comes an expert with first hand
experience to give you real data!