There are often controllers which do indeed just mimic the signals. Doesn't work with every appliance, it depends on the way it's implement and if the manufacturer wanted to make that approach infeasible.

But there absolutely are options to record such Signals and then replicate them via home assistant - I used them before to control a ceiling fan and various infrared devices (same idea, but not a radio there instead a "blaster" - I think it was called)

I didn't set it up again after my last move though, as I couldn't mount the ceiling fan in this apartment and the Infrared devices were just my media center (tv, audio), which are hardly in use currently

The key with all furnace/heating/cooling automations is to start at the source of heat - figure out what IT needs to do what it wants, and work from there.

They're almost always incredibly simple at the furnace/boiler - you just need to make sure that you never turn the heat on without the pump/blower or whatever is required.

My complicated Eco controller ends up with three outputs: blower on, heat on, cooling on. Three wires.

Take a look at SwitchBot. They have a device that can tap buttons to solve these kinds of problems. They also have a device for tilting blinds by twisting the rod, which could maybe be modified to twist a knob.

If it's actually using radio, almost certainly there is a radio receiver unit at your furnace which converts the radio commands to simple voltages on physical wires, likely to power a 24v solenoid. All you need to do is hook in a esp32 or similar to also send those voltages when it receives a command.

I know some cabins are quite remote but do you have a trusted neighbour who would do it for a case of beer?

My mum's neighbours buy milk and bread and turn the heating on! I don't quite trust my own neighbour to do that but it's awesome for her