Thanks Zoe
Here's a link to the photos, but you'll need some context first:
I have included a whole bunch of photos of wires in the boiler because I don't know which ones are relevant. I actually had professionals come in to get the thermostat installed, and they spent about 2 hours trying to hook it up and figure out why it wasn't working right, and eventually gave up. So they're the ones who hooked everything up. You will notice the picture of the thermostat wires is currently hooked up with only the W and R wires connected. The C wire is tucked in behind and is not used because this thermostat uses batteries. But it's just the W, R, and C wires that were being connected to the Google Nest Thermostat.
The people who worked on it said they had never installed a Google smart thermostat before, they only dealt with Honeywell products, so they were a bit like a fish out of water. Since the wires at the thermostat look super old and I figured maybe that was the problem, I actually asked the guy if he could connect the Nest Thermostat directly to the boiler, without using the wires that are run throughout the house. So he tried that, and it had the same result. The fan would just automatically turn on no matter what, and kick off a minute later, only to immediately start back up again. It seemed the issue was with the power draw from the C wire, as though the request for power to the boiler would trigger the initial startup phase of the boiler, but then stop immediately, leaving the fan to run on a 1 minute cooldown timer before shutting off, only to be requested to turn on again.
So I know the issue is not the wires themselves in the house, just so you don't think that's the issue. I included the pic of the old thermostat connection but it really isn't relevant. What matters is what's at the boiler, because we couldn't even get the thing to work when we connected it directly.
Hope that helps clarify.
