Can you Manually copy User Folder to new Mac?
If the drive is fine and only the old Mac broken, then you have several options. These are the steps I would recommend:

With the drive attached to your new Mac, open disk utility, hit view and select show all devices. Then choose the parent volume of the drive and click first aid. Do the same with its volumes. If an error occurs try again multiple times and/or in Recovery Mode.
Try simply booting from this drive. Shut down the Mac. Disconnect everything, then only reconnect the old system drive. Start your Mac and hold the opt. If the boot loader detects your old system drive to be bootable it show up and you can select it to boot from that drive. You can then do a TM-Backup or even continue using it that way if you want.
As already suggested by @nohillside you can also try using Migration Assistant.
If you want to completely control what files will be on your new Mac, then I suggest only copying specific files from your old drive to the new Mac (e.g. only the Home folder and no Apps).



To restore your user account I would recommend creating a TM backup, since this will also restore settings and all programs for that user account.
If you only want to restore the data of your user account, it is
perfectly fine to just copy the contents of the Home folder (e.g. the
Movies folder).