cannot enter time machine

Just so that everyone is on the same page, we specify and track backups with:

System Preference > Time Machine

and we Examine and Manipulate and Restore by "Enter Time Machine" using:

Time Machine.app ... available only in the /Applications folder.

If the Icon on the menuBar does not work, that is lamentable, but not deadly if you can double-click on the Icon in /Applications.

Time Machine backup is an integral part of MacOS. It needs to be, so that it can have access to backup everything on your Mac. (Including System Files and Multiple Users if you have created Multiple Users.)

It is odd that Time Machine.app cannot be launched. Use of Finder will not suffice, because most references to files in a particular Backup (as of a specific date) are done with Hard Links, and those are invisible to Finder, even if you "show all files".

You CAN do a Restore of the latest version using Recovery HD, or using Internet Recovery, or by booting from the tiny EFI file added to backup drives after 10.7.3.

If a Restart does not fix the underlying problem, you may need to do a MacOS Re-install-in-Place to get it back again. By design, this only re-writes System Directories, and leaves your added Applications and files untouched. However, if the underlying issue is a failing drive, re-writing the 350,000 files that comprise MacOS may push it over the edge into failure.

If you can't back up or restore your Mac using Time Machine - Apple Support

You should just be able to click on Time Machine icon in your Dock, or double-click TM icon in Applications?

Wow. That is really Ironic that the thing that absolutely refused to run was Time Machine (Restore).

Great work on getting it sorted out!