Take it to a different dealership and don't mention anything in particular. Pay the $150 diagnostic fee and have them inspect it. Take the findings to your original dealership and have they fix it and reimburse you (if they did screw up)
My 2016 mounts have started to crack and the movement seems to be the same like yours. However, no weird sound. It's hard to diagnose when you're driving vs stationary.
If it's leaking coolant, you'll likely see white residue after it's evaporated. You said theres smoke. Put the car in park, pop the hood again, and have someone rev the engine. See if that would generate more smoke.
All in, I don't think Mazda went above and beyond. However, some help is better than no help at all. At this rate, the OEM route is definitely a better choice. They do need to program the new diff in so I'm not sure if outside shops have this capability.
I would get something with lower miles if budget isn't the main factor. Lots can go wrong (mechanically and physically) as parts will deteriorate due to age. Have a mechanic that you trust go over the condition but at that age and miles, it's a gamble imo
So what's the breakdown and how much is Mazda covering?
https://mazda.oempartsonline.com/oem-parts/mazda-differential-assembly-kat227020?c=Zz1zdXNwZW5zaW9uJnM9YXhsZS1hbmQtZGlmZmVyZW50aWFsJmw9MyZuPVNlYXJjaCBSZXN1bHRzJmE9bWF6ZGEmbz1jeC05Jnk9MjAxOCZ0PXNpZ25hdHVyZSZlPTItNWwtbDQtZ2Fz
Diff is $2k
So...
Happened to my 2016 once. I was also getting on the freeway and it went into limp mode. Turned off and turned on and self fixed. Everything been fine so far. Knock on wood.