Does erasing the HD and then restoring from time machine backup remove applications?

Solution 1:

This is an answer to the question as asked, but I suspect further investigation will be necessary.

Reinstalling the OS without removing existing detritus will quite likely return you to the exact same state.

Reinstall from Recovery without erase will clean up the OS itself without affecting any other data/apps. If the OS itself was at issue, then it will fix the OS. It will not fix anything user-installed that is slowing the OS down.

Erasing then installing from scratch then migrating in from Time Machine will return you to the Mac as you last had it. No change, except the OS cleanup from the previous step.