Unfortunately, Bogdan, there isn’t a simple answer. Neither the Moxiecode or the FCKEditor development teams have their editors set to automatically cleanup Word HTML. Both provide plugins for this but, as you have found out, in order for them to work properly you have to initiate the plugin via a "Paste from Word" button. Neither will automatically do it via a CTRL+V (CMD+V on Mac) so removing the standard Paste button doesn’t fix the problem entirely. Question is, if the Word cleanup plugins were on all the time would they mess up pasted code that isn’t specifically Word HTML. That’s probably one of the primary reasons why they don’t make it a default cleanup option and have implemented it as a stand-alone plugin.
Now, in regards to your users not using the "Paste from Word" feature properly...well...my take on that is this...
You have to train your clients on how to use MODx anyways. If this is one of the features you emphasize to them and tell them how important it is to use and they don’t use it right, whose fault is that? I tell users around my office all the time and harp on them about a variety of things like this. You can only do so much. It’s their choice on whether they listen and learn or not. As long as you teach it to them and emphasize the importance then that is all you can do.
Also, if you really wanna bitch about it, blame Microsoft!

I’ve always hated the fact that they use such crappy markup in the HTML of pasted text from Word! Good news is that Office 2007 seems to not have this problem nearly as much. About the only thing I noticed when pasting from Word 2007 is a few class attributes.
Jeff