Magic Mouse Model: No.:A1296 3vdc not moving cursor

Ok, set aside what you can do with clicking. Left or right, they perform actions defined in Settings.

Moving the cursor is a primary function, and if that doesn't work then the mouse is probably defective. You alluded to the fact it didn't work on another Mac, which supports that conclusion.

Buy a new one.

You need a newer Magic Mouse 2.

It should work. I have one and it works with that model MBA. It works with almost all Macs since Leopard, at least.

Make sure you're using good quality alkaline batteries. Rechargeable ones won't work. And make sure they're inserted correctly. Both + near the LED, not opposite one another. Otherwise you'll get the difference in voltage which will be near zero.

Not sure what you mean by a right "button" or how you intend to scroll with it. The MM is only clickable, though you can click either side due to its touch interface. Technically I suppose it's a "button" but there is only one. And you still can't scroll with it.

The italicized text is what doesn't make sense to me:

Behavior: It turns on, it's lights, recognize and connects to the bluetooth, I can scroll and click with the right button.

I can't: - Move the cursor or click with the right button.

Hence my confusion.

FWIW, this is not the first report of problems with the battery powered Mac Mouse and newer versions of macOS.

I only used it long enough to confirm it still works, which it did. Batteries were new of course.

The "right button" comments just don't make sense to me.

I still have an older Magic Mouse. I'll check it with my M1 Air running Sonoma when I get a chance. Maybe the OP just meant right click.

Yes, I know. :)