Hi, thanks for your response. I'll try to respond step by step here to make clear what steps I've taken and the status of each issue.
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Re: "When I turn on Find My I get "Find My Unavailable, Unable to connect to server.
Please try again later."
Sometimes one of the various Apple services, like "Find My", is unavailable for a short time.
Eg: There may be an outage. You can easily check out whether this is the case by going to the website below. Services marked with a green dot are available ... (it says "all" are available for my location today) Check the System Status Apple - Support - System Status page to see if there are any outages.
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At the same time that my Mac was showing Find My as Unavailable , it was available and working on phone and iPad. I have just now checked the website you provided me (thanks!) and it shows Find My as available and I still have the same situation, Find My showing as unavailable on Mac, working perfectly on other two devices.
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Re: "Find My on iPhone list of devices shows the phone and iPad, but still shows my old MacBook Pro instead of new MacBook Air".
Also: "Apple ID list correctly shows my iPhone, iPad, and this new MacBook Air (but shows "find my" off on MacBook Air. It's "on", but, as I said, shows "cannot connect to server."
As both iPhone and Apple ID do not show Find My on for your MacBook Air:
If you want to check the Find My set-up for your MacBook Air, you might want to scroll down to: "How to turn on Find My for your Mac" in Apple Support's article: Set up Find My - Apple Support
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Yes, I did these steps when I set it up. According to this, Find My Mac and Find My Network are both on. (I just rechecked, they still show as on. When I try to use Find My it still shows Unavailable.
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If needed: Trouble-shooting "Learn what to do if you don't see your device listed":
See: If Find My is offline or not working - Apple Support
Yes, I have been through the steps on this page.
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Re: "Bluetooth - phone and iPad both recognize and MacBook Air.
MacBook Air doesn't recognize any other devices. Air bluetooth is on."
With your MacBook Air and devices all signed in to the same Apple ID:
Sign in with your Apple ID - Apple Support
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This seems to be instructions for signing in on various devices. When I'm already signed in on all devices (which I am), I don't see how I can sign in without signing out first (which I've considered on the Mac, but so far hesitated to do because that would mean I would have to set up my Apple Pay/Cards again.) I did sign in to iCloud.com and could see all of my apps. When I went to Find My on iCloud.com it showed the phone and ipad only, just like I'm seeing everywhere else.
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You may want to use : macOS - Continuity - Apple
Eg: Use Continuity to connect your Mac, iPhone, iPad, and Apple Watch - Apple Support
and
How to: Use Continuity to work across Apple devices - Apple Support
I have read through all of this material. Don't immediately see how it can apply to my issue.
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For using Bluetooth devices on your Mac:
Pair devices one by one: a device can only be paired to one other device at a time.
The device to be paired with needs to be discoverable … in discovery mode … may need access to power, or be charged : Connect a Bluetooth device with your Mac - Apple Support
I have put all three devices in discovery mode, all either connected to power or fully charged. Mac doesn't discover the other two devices.
I don't have a device I need to pair with the Mac at the moment, just doesn't seem right that the Mac doesn't discover the phone or iPad while they both discover the Mac.
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Thanks again for all of this information. It's not like I have ever needed to use Find My, but it bugs me that it's not working on the Mac.
Weirdly, Air Drop works between all three devices. (Maybe not weird, I don't really know how Air Drop works.)