The transition between Ancestry's older Online Family Tree system and the current system with Ancestry Member trees took place in 2007, according to this post on the Ancestry blog: Online Family Tree Announcement. I did not use the old system, so I do not remember if you could download GEDCOMs from other people's trees there.
There are other products associated with or purchased by Ancestry which might be confused with this: RootsWeb's World Connect Project (link to their FAQs via Wayback Machine, since RootWeb is currently offline), Ancestry's OneWorldTree (the linked page says this was discontinued in 2013), and Family Tree Maker's World Family Tree CDs.
Rick Crume's (premium) article at the website of Family Tree Magazine (originally published in the August 2000 print issue of Family Tree Magazine), Families on File gives an overview of what searchable databases existed at that time. (The site offers a limited number of free views of their premium articles per month.)
The timeline, according to Crume's article:
- WORLD FAMILY TREE : CD-ROMS, introduced 1995
- ANCESTRY WORLD TREE : Created in 1997
- WORLDCONNECT PROJECT : introduced in October 1999 (as far as I can remember, this did allow some GEDCOM downloads)
This screenshot is taken from Google's cached copy of a support article Uploading and Downloading Trees (presumably from Nov 14, 2016) which has been moved or removed:
Only the owner of a family tree can create and download a GEDCOM file of the tree; people invited to the tree as a guest, contributor, or editor are not able to download a GEDCOM.
The article refers the user to this YouTube video: How To Create and Download a GEDCOM | Ancestry Academy
For an overview of the company history and a list of past products, see Wikipedia's article Ancestry.com.