I remember reading something years back that said when Sky came along, they were looking for a more professional, polished image for their 'brand' - which probably explains Jamie Carragher.
I remember Saint and Greavsie - wasn't old enough to know who they actually were but enjoyed the light hearted banter at a time where football was a serious business. As you say, I'm sure they've had a big influence on a lot of today's programming style.
Just googled and Greave's isn't in the best of health (stroke a few years back, confined to a wheelchair) but saint is still going strong and attending matches.