Yep! A very common issue! Your hitting the HFS+ Vs APFS problem! The macOS you are using is newer than what Apple offers you via Internet Recovery! To compound this you maybe using the wrong SSD.
Using a second Mac (or reinstall the original drive if you can get it working) prep up a bootable OS installer USB thumb drive with Big Sur.
Start off by formatting the thumb drive with a GUID Mac format. Then download the installer from here [https://support.apple.com/en-us/HT211683|How to download and install macOS] then follow this to guide [https://support.apple.com/en-us/HT201372|Create a bootable installer for macOS].
Now use it to boot up your system you should now be able to see the internal drive using Disk Utility from the thumb drive.
If not you’ll need to dig into why. Are use using the correct SSD? Are you trying to install a M.2 SSD via an adapter? Is the system damaged in some way?
If you do see it again format it and install the OS that should do it!