I just ran into this issue with my ecobee 4 thermostat & eero Pro system with HomeKit enabled and Accessory Security set to Auto for ecobee Inc. Thermostat in the Home app. I was experiencing the issue where it would connect to my wifi network, but not ever connect to ecobee servers. I went through every possible troubleshooting step on my own, and then with ecobee support, whose did show that it could connect to their servers when connected using my phone's hotspot, isolating the issue to my eero Pro network. Their proposed solution was to completely reset my wifi network (uh, no). After they said "Contact eero or your ISP," I did some more digging on ecobee's wifi settings, and it looked like they recommend allowing two ports in your eero settings. Here's how to do that:
Go into the eero app, tap Settings at the bottom, then Network settings, then Reservations and port forwarding. Tap "Add a firewall rule" under the IPv6 Firewall Rules heading. Select your thermostat from the list of devices currently on your network. Mine was named "Living Room," and once you've selected it, tap "Add a firewall rule" at the bottom. Then, add ports 8089 & 8190 one at a time to the box. Here's the documentation from ecobee:
https://support.ecobee.com/hc/en-us/articles/227873287-Connecting-to-the-Internet
Hopefully that helps everyone else! Feel free to ask any questions if needed!