I was having the same issue and nothing seemed to work. It was my son's Apple Watch SE and I was trying to set it up 'for a family member'. I'm on an iPhone 15 Pro with the latest iOS 17.3. Whenever it would get to the step where I would want to select his account, it would only flash and then start unpairing.
After many futile attempts of resetting the watch and doing everything short of completely resetting my phone, I finally got a support tech on the phone that pointed me in the right direction. He suggested it may be my work's management profile that might be blocking the setup and causing the watch to unpair at that step (for anyone looking, you can find that setting under Settings > General > VPN & Device Management). I removed the management profile but that actually didn't fix the issue. It seems that even the profile having been on there leaves something lingering that was causing it to block that particular setup of the watch.
So the only logical step was to try a phone which didn't have a management profile (ever). Fortunately, we were able to use his mom's phone and ran through the setup without any issues.
Not sure if this is exactly the issue that others are having, but it was the only way we were able to get it working so thought I would share. And while I'm not usually the type to write on these boards, this process had me losing my mind and tearing my hair out - it was not fun; so another reason to share maybe a path for someone out there with a similar issue to get their watch working. Hope this helps.