I believe the problem here is that "The Sims 3 Mod Manager" was a tool that attempted to automate the installation of mods the old way. That is, from before Patch 1.12, the one that came out with Ambitions. It has therefore altered the TS3 files and file structure within Programs, not the game folder as we know it today in Documents. Back in 2009 when this tool was released, that was how mods were handled and it hasn't been a supported configuration since around five years ago.
Hate to say this, but I don't think there is a shortcut that can be taken in this case outside of a total uninstall, registry clean, and reinstall of everything. Just having removed the Mod Manager would not have put things back the way they were.
http://simswiki.info/wiki.php?title=Game_Help:TS3_Uninstall
Patches, by the way, do not have be installed separately.
http://help.ea.com/en/article/the-sims-3-super-patcher