Nest devices do integrate, sort-of, with Home Assistant. But, the process is to create an "app" inside the Nest + Google home automation ecosystem. It is an absolute mess and there is nothing to be gained by going into that Mordor. I tried to get it working, and gave up, but I would much rather use a device that is "hackable." That device you mentioned looks great, and does not appear to be available outside of Ireland, as you noted. But, I'm going to use that as an inspiration point and see if I can find something similar. Maybe that manufacturer has awareness of a US-based device provider, I'll contact them.
Really appreciate you sharing this work, really fun stuff!
If you're in north America (probably elsewhere, but I can only say for NA) then changing the thermostat is super simple to do. If your system is 24v, which is very likely, then swapping the nest device for something better isn't too complicated. I can't provide a brand recommendation, but I'd find a thermostat that supports homeassistant and has a reverse engineered API available.
Directly controlling the HVAC with a relay board would also be very doable, but is a more involved project.
I installed my nest, and it was indeed very simple. And, it appeared to work with Home Assistant, but like I said, the devil is in the details. It was a very frustrating experience.
The problem is that if I look here I cannot tell the first thing about which device is supported and how well. HA does not have a great filtering system.
https://www.home-assistant.io/integrations/#climate