So full disclaimer I'm no expert either, discussing with HVAC contractor is good idea. It doesn't look like you have a vendor specific communicating thermostat. The equipment interface model you mention simply appears to be solution for being able to leverage a 3 wire tstat cable. Most other thermostats have similar "equipment side" devices to support the older cable. Of course if possible getting modern 5 wire cable expands options.
As far as tstat keeping itself pegged EXACTLY. I run my system in auto so have a heat setpoint and cool setpoint. By default they're 5 degrees apart but ecobee allows as low as 2 degree difference to avoid oscillating back and forth heating and cooling unneccessarily. Can also adjust how quickly system kicks on for both heat/cool in advanced settings. However looking at your honeywell thermostat it appears they too have some remote sensors. I assume they work the same, but not sure if Flair integration works with Honeywell. I can confirm Flair has pulled in my ecobee remote sensors.
The reason I suppose I prefer thermostat specific sensors rather than pucks informing flair and flair driving the tstat is for KISS principles. Letting the thermostat do what it is best at (deciding when to run and not run heat/cooling) while letting flair do what its best at (determining when rooms have reached desired settings and adjusting vents accordingly) seems to minimize requirement for both systems to work well and play nice.
Flair is new and while their logic may ultimately prove incredibly intelligent and reliable. In current state, I would not let their pucks drive my thermostat. I think Flair likely feels similarly at this point due to them disabling the automation portion of their suite for the time being...Further, even once everything is working as intended using pucks to drive thermostat adds system dependence on flair device internet connection, flair cloud, flair <-> tstat vendor api, tstat vendor cloud, tstat device internet...Any issue in that chain and your system isn't working as intended. Using the native tstat and remote sensors to drive tstat generally means you can lose internet all together and be fine....
Basically in the setup of a smart tstat with sensors, the thermostat alone will try to balance your system out, meaning your downstairs will be little bit below your setpoint and your upstairs a little bit above. Flair gives that balancing a boost by controlling vents based on the values reported by the remote sensors. That of course helps hasten the distribution of air where needed which will in turn get your sensors in those rooms to report back to your tstat that they're at the desired level sooner.
Hope that makes sense.