Hi Zingno,
If you haven't done so yet, restart your iPhone. Although simple, this step can help correct some unexpected issues. It's also a good idea to make sure your iPhone is running the latest iOS version, since this can help with overall performance: Update your iPhone or iPad
If an update is available, be sure to create a backup before installing it.
You mentioned following steps from an Apple Support page, but if the behavior continues, these steps would be the best to follow to resolve the issue, including resetting the AirTag as a last step: Set up your AirTag using your iPhone, iPad, or iPod touch
Keep in mind that resetting your AirTags will likely remove the other Apple IDs that it's shared with.
If you can't set up your AirTag
1. Make sure that your device is ready for setup.
2. If the setup animation disappeared, press the side button or Sleep/Wake button on your iPhone to put your device to sleep. Then wake and unlock your iPhone. Wait up to 15 seconds, and the animation should reappear.
3. If you have multiple AirTags to set up, make sure that only one is near your device at one time.
4. Remove and then replace the battery in your AirTag.*
5. If your AirTag still won't connect to your device, reset your AirTag.
If needed, here are the steps for sharing an AirTag: Share an AirTag or other item in Find My on iPhone
Regards.