Well, define "support."
As a preliminary matter, we don't know who wrote the FAQ. We don't know who at Hasbro (if anyone at Hasbro) authorized the FAQ. The FAQ is what is considered extrinsic to the OGL and is post hoc (after the faq heh ... fact,/event).
You ask why they would "support" the OGL if they never wanted it? Well, look at the timeline!
In 2000, the OGL was published.
Until the latter part of 2001, WoTC was independent (autonomous). Then they were folded into Hasbro. It usually takes time for the higher-ups to start paying attention.
In 2003, they did the first distancing from the OGL (and 3e) be releasing 3.5e.
By 2007 they had already announced the development of (non-OGL) 4e. Remeber that 4e is when Hasbro put a bunch of resources into D&D.
In other words, after a relatively stable edition from 1974-1999 (OD&D-2e were largely compatible), they released a half edition within three years, and a fill new edition within seven just to try and get away from the OGL.
Hasbro has never supported the OGL. People within the subsidiary, WoTC, have.
That's why I say that things are a happy accident. Once it was out in the wild, it was hard to contain. I would guess that we have gotten to the point that Hasbro thinks it is worth the PR and legal fees to try and put the genie back in the bottle. Who knows?