How do I perfectly center a table on the page in Microsoft Word?

This should be pretty easy using the Table's properties. Try this:

1. Select your table, then right click.
2. On the Table Tab, Text Wrapping, choose Around. This should activate the Positioning option.
3. Select Positioning, and in the Table Positioning dialog box, set:
   • Horizontal Position = Center, Relative to Page
   • Vertical Position = Center, Relative to Page
   • Move with Text = unchecked
4. OK out of the dialog boxes and your Table should be centered. And it should stay centered regardless of changes to the Table (e.g. adding rows) or changes to surrounding text (including the extra, "silly line".

FWIW, you shouldn't even have to modify the Page Layout Property, this solution addresses the Table's location regardless of the Page's properties, so any text you may want to add could still be top justified.

Horizontal centering is fine as you stated, but IMO to achieve perfect vertical centering you'll have to place the table inside a text box, resize the text box to have the same margins on all 4 sides of the table, remove the text box border and finally center the text box on the page using either
Format / Position or Format / Align: