I wonder if the low altitude warning from ATC was their first notice that they were off glideslope. Note that the descent angle was increasing over the 3 nm (or roughly 1 minute) prior to the 4.8 nm data point. The radio call from the aircraft in response to ATC's warning came 20-ish seconds after the warning was issued, and didn't sound real confident and certain - as if they were still processing the warning.
The equipment at SFO definitely has newer then anywhere I've worked but where I do work the radar functions this way: It has database the terrain mapped out and it projects the position and altitude of an aircraft based on their descent rate and ground speed in reference to the terrain. Based on it's calculations it decides to activate the MSAW (minimum safe altitude warning). There is a box programed around the approach path that is used in the calculations, but predominately it's the projection of the aircraft's position base on speed and descent rate. I got an an MSAW on an aircraft descending out of FL190 over water. It was an F18 descending at close to 10k fpm and 450kts ground speed. The radar determined in x number of seconds if that trend continued that it would be in unsafe proximity to terrain.
On your chart you can see an increase in their descent rate between 7 and 6 miles (compared to the trend from 10+ to 7 miles) this corresponded with a sharp increase in groundspeed. I'd say this is what triggered the low altitude alert. The relationship to the glideslope was probably secondary in that it only meant they weren't very high and didn't have a lot of time to make up for a loss of altitude especially considering the ILS is still OTS at SFO as far as I know.