"Is there a way for me to retrieve the data from my nonresponsive 2017 Macbook Pro's SSD, which is soldered to the logic board, if the repairer can't repair it and has offered me a replacement device? Can I use Time Machine to transfer the data to the replacement device? And if so, how would I go about doing that?"
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Troubleshooting a Corrupt Hard Drive:
A. Hold It!!! — Recover this Data!
Why risk lost data? Rather than erasing the hard drive. Take this into your local AASP (Apple Authorized Service Provider). See what services they have to offer.
B. Go Here: Find Locations - Apple Authorized Reseller. Click: Service & Support > Enter your location information, and see if there is an AASP nearby. Contact one that shows up to find out more about the services that they offer to fix this.
C. Migration Assistant:
Likely, they’d go about using Migration Assistant. See if recovering this hard drive’s data is a possibility of a service they offer. If so, an AASP might do so, and then would use Migration Assistant to move forward with this and transfer it all over. Go Here: Migration Assistant - Apple Support