I not sure what a gravity hot water pump system is, but yes it does have a pump and it normally copes fine. It seems to be working and I am just coming to the end of my round of bleeding the rads. I discovered that the lock valve on one seems to have been nearly closed. I don't know if you have been following this saga, but I mentioned that of the 14 rads in the whole system we only use 7 so the strain on the pump ought to be less than it was designed to take. But I have to admit that the last plumber to look at it in any detail, (he was replacing the tank) said that the pump wasn't really rated as powerful enough for the system. Without being asked he fitted gate valves either side of it to make replacing it easier than it otherwise would have been.
He is a decent bloke, he was the plumber who did all the plumbing for our ex-son-in-law and his father who built houses for a living so we trust him implicitly. Sadly he has retired and his son, who is supposed to be doing the same job, is useless and his father is the first to agree with this.
Is the "gravity" part of the system the fact that it has a small header tank for the CH and a much bigger one for the hot and cold water for most of the house?
I have come across a system that relied totally on gravity and I think the fact that hot water rises. It was in a small two bed terraced house. It always seemed cold to me but maybe the burd who owned it didn't feel the cold!
The pump is on the first floor, next to the hot water tank.