Simple answer is a very big '''YES!''' To your title question: Can I restore from a time machine backup after doing a clean install?
'''Now the long full answer!'''
You are not going to test the drive cable with the setup you are proposing. Lets be clear, your drive cable is likely failing more so than the SSD.
Here’s a bit more on the subject [https://beetstech.com/blog/prevent-a1278-mid-2012-hard-drive-cable-failure|Your Hard Drive Cable Is A Ticking Time Bomb]
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So… You really need to replace the HD SATA cable [https://www.ifixit.com/Store/Mac/MacBook-Pro-13-Inch-Unibody-Mid-2012-Hard-Drive-Cable/IF163-041?o=1] and yes! This is the 2012 version which is the better cable for SSD’s and here’s the guide to put it in [https://www.ifixit.com/Guide/MacBook+Pro+13-Inch+Unibody+Early+2011+Hard+Drive+Cable+Replacement/5120]
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So… You really need to replace the HD SATA cable [product|IF163-041] and yes! This is the 2012 version which is the better cable for SSD’s and here’s the guide to put it in [guide|5120]
As far as backups you don’t need to touch your external drive TimeMachine at all here! All you need is a SATA to USB adapter like this one [https://www.startech.com/en-us/hdd/usb3s2sat3cb|StarTech 2.5" SATA to USB adapter cable] to connect your original drive and then use the Migration Assistant app tp transfer your user accounts, apps and data to your replacement drive.
I strongly recommend you stick with Sierra as the newer macOS’s don’t run very well under SATA drives.
You are not going to test the drive cable with the setup you are proposing. Lets be clear, your drive cable is likely failing more so than the SSD.
Here’s a bit more on the subject [https://beetstech.com/blog/prevent-a1278-mid-2012-hard-drive-cable-failure|Your Hard Drive Cable Is A Ticking Time Bomb]
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So… You really need to replace the HD SATA cable
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So… You really need to replace the HD SATA cable [https://www.ifixit.com/Store/Mac/MacBook-Pro-13-Inch-Unibody-Mid-2012-Hard-Drive-Cable/IF163-041?o=1] and yes! This is the 2012 version which is the better cable for SSD’s and here’s the guide to put it in [https://www.ifixit.com/Guide/MacBook+Pro+13-Inch+Unibody+Early+2011+Hard+Drive+Cable+Replacement/5120]
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As far as backups you don’t need to touch your external drive TimeMachine at all here! All you need is a SATA to USB adapter like this one [https://www.startech.com/en-us/hdd/usb3s2sat3cb|StarTech 2.5" SATA to USB adapter cable] to connect your original drive and then use the Migration Assistant app tp transfer your user accounts, apps and data to your replacement drive.
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I strongly recommend you stick with Sierra as the newer macOS’s don’t run very well under SATA drives.
You are not going to test the drive cable with the setup you are proposing. Lets be clear, your drive cable is likely failing more so than the SSD.
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Here’s a bit more on the subject [https://beetstech.com/blog/prevent-a1278-mid-2012-hard-drive-cable-failure|Your Hard Drive Cable Is A Ticking Time Bomb]