Will Apple's Data Recovery Service successfully retrieve files from an iMac High Sierra with startup issues?

Back in December, I discovered that my iMac High Sierra had a problem where it started to turn on, but then halfway through startup, it suddenly turned off by itself, but still booted in Recovery Mode and Target Disk Mode. My Mom and I took it to The Geek Squad at Best Buy last month and they said the problem had something to do with the hardware, but they were nice enough to agree to try and back up my files and data from that computer with a Seagate External Hard Drive that I had. When they were unable to do so, they agreed to send it to Apple to recover my files via a professional data recovery service, though they said it might take about a few months for them to do it.

I don’t know for sure if my iMac High Sierra’s hard drive is severely damaged or not, but at this point, I’d do anything to save all my files from being lost. It almost happened to me once before, so I’m hoping I’ll be lucky again this time. Do you think there’s a chance the data recovery will be successful?

[Re-Titled by Moderator]