The military has airborne and surface-based electronic warfare platforms that can be used to jam GPS. That's what is referenced on that page.
The FAA is not considered an authority for GPS, with the exception of certifying that WAAS is in-service for aircraft when it is meeting FAA specifications. For civilian information purposes the appropriate agency is the US Coast Guard NAVCEN.
The Air Force Space Command is the holds primary developmental, specification, and operational responsibility.
But I've searched around and can't find anything that indicates that this is a feature of the satellites themselves, rather it's just as you say, the US military has replaced SA with the general ability to jam local radio traffic, including GPS. E.g. [1] is another source for that.
So it does look like the US can't block GPS for entire countries anymore, I take that back.
I must say though, I think any public commentary about GPS capabilities should be taken with a huge grain of salt. It's a military system, they're very likely lying or not telling the full truth about what it's really capable of. E.g. I find the claim that the new generation can't do SA by design dubious, surely just fuzzing the signal by some offset is a rather trivial change.
1. https://web.archive.org/web/20050921115614/http://ngs.woc.no...
[1] http://www.dote.osd.mil/pub/reports/fy2011/pdf/af/2011gps.pd...
Each of those satellites in view would have to be implementing this scheme you suggest in order for it to be effective.
Since the satellites only have a single transmitter for each signal their entire receiving area would be affected by whatever modifications are made to implement SA. This means that to turn SA on for a single area more than half the globe would be getting a degraded signal from 1 or more satellites, which ones being undetectable.
I'm afraid SA is an all-or-nothing idea. Area denial (jamming) by surface and airborne transmitters is a much more workable approach. One could imagine jamming from space, but not using the GPS satellite vehicles.
So, to GPS. In the original GPS SA it is certainly true that many satellites would have to be using degraded mode to guarantee degradation for a specific point. However, the newer satellites have an additional directional antenna. While primarily for transmitting M-code to a specific region, it could equally be used for transmitting a self-jamming signal.
There is an enhanced footprint signal boost capability for the military but that's the opposite of what you describe.