I formatted this hard drive at least twice and used it for a few years. I thought I had lost it, but when I moved I found it stored in a box.
I tried searching for the hex string "fd1701308201130201010420'', but no luck, I don't know if this prefix covers wallets from early 2010 or if all wallets contain the prefix 04 20 preceding a private key.
Formatting these days usually only means erasing old and writing new partitioning and filesystem data structures. In best cases it doesn't overwrite file contents.
But chances diminish when you say you used that drive after formatting. Using is means overwriting old content's sectors with new content which makes old overwritten content unrecoverable. It heavily depends how much new content was put on the drive after formatting.
And the ususal best practices when data recovery is performed: never work on the original data and storage media. Make a master forensic bit-by-bit copy, every sector of the source drive needs to be copied! If data content is really valuable, you make at least two redundant master copies.
From a master copy you make one or more working copies. You only work on the working copies!