Bluetooth pairing
Trying to pair iPhone with iPad but other devices gear never stops spinning and devices will show up temporarily and then disappear
iPad Pro, iPadOS 16
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See the information in: Bluetooth: Why can't I pair my iPhone or … - Apple Community
If you are attempting to use Handoff/Continuity or AirDrop, that does involve Bluetooth but the devices are NOT paired; Bluetooth just needs to be on and the devices in range of each other. The following may help with Handoff/Continuity issues: Use Continuity to connect your Mac, iPhone, iPad, and Apple Watch - Apple Support
Also see Share files between your Mac and Bluetooth devices - Apple Support