Sorry just to confirm, you can reinstall just parts of your file system from a time machine back up?
Yes. Ideally if you have all the applications (and any product keys) on hand you wouldn't want to restore the application folders. You'd be better just restoring your personal files and doing a clean install of the OS and any applications you want to keep.
There's some stuff on this page, it's a little out of date but not massively, covering using migration assistant to selectively grab stuff from a time machine.
http://www.macworld.com/article/1165784/how_to_restore_data_from_time_machine.html
Good stuff here: http://pondini.org/OSX/SetupLion.html
Here's a quite short version of above: http://www.macworld.com/article/2066996/how-to-transfer-backups-from-time-machine-to-a-new-mac.html
If it were me, I'd only restore/migrate the "User Account" data, not Settings or anything else as that's where any "slowness" may be "hiding".