I have been encountering the same issue with my Pixel 8 for the last couple of days. I think it has only started since it updated to an Android 15 build.
I'm not using multiple profiles, only the owner profile.
Private DNS is set to automatic. My firewall blocks outgoing DNS-over-TLS but offers itself as a local network resolver via DHCPv4 and IPv6 RA. DNS-over-HTTPS isn't blocked by the firewall, yet. I'm not certain what Android will resort to using under those conditions.
The update check has failed four times during the last 24 hours according to Notification history, but I think there were a few before that as well. It's certainly happening far more frequently than it has in the past. It may have happened occasionally before, but only as isolated occurrences that weren't worth remembering.
I'm using DuckDuckGo browser's App Tracking Protection, which sets up a fake VPN connection to filter outgoing traffic from the tracking & telemetry frameworks built into apps. I don't believe has caused any DNS-related issues before, but I'll try disabling it to see if that makes any difference.