AutoComplete does work for all Building Blocks including AutoText in Word versions beginning with Word 2013.

The screenshots above are from Word 2021. The one for the date will not appear until a day is typed unless you capitalize the month name yourself (not using AutoCorrect).
I believe you are talking about Word 2003 which was Word 11, I believe.
If you know the name of an AutoText and start typing if it does not pop up, the F3 key should trigger it s well. *(*Fn+F3 on many computers)
Automated Boilerplate Using Microsoft Word
However, AutoComplete for Building Blocks and AutoText will not show up immediately after you start Word.
In my experience, the first use of a Building Block/AutoText loads them into memory. Calling up the Building Blocks Organizer under Quick Parts would also constitute a first use. This may be what you are experiencing.
History: MS did away with AutoText and AutoComplete in Word 2007, restored AutoText and partially restored AutoComplete for Building Blocks stored in the Normal template in Word 2010, and fully restored it in later versions.
AutoText, Building Blocks and AutoComplete for them are not functional for templates that are stored in SharePoint.
My tests are done on Word for Windows 2021 and 365 run on Windows 11 and Windows 10, respectively.