That's a pile o' pumps!
If you do that it's more than a simple swap out with the zone valves. Plumb it primary/secondary, hydraulically separated and move the expansion tank to the input side of the old pump, using the old pump on the boiler loop, and use check valves on the zones (one per pump) to keep the pressure difference of the active zones from creating a reverse flow on those not calling for heat. (If every zone has quite a bit of head the reverse flow may be slow enough to not much matter in practical terms.)
(Ignore the fact that this diagram is pumping toward the boiler as opposed to away, and move the expansion tank to the other side of the separator.)
The hydraulic separator need not be a purchased component- it can be fabricated with short length of pipe with closely spaced Tees if you do it right. Alternatively you could use a buffer tank as a hydraulic separator to add mass and inhibit short-cycling on single-zone calls for heat, etc. In some instances it makes sense to use a "reverse indirect" type buffer tank & hydraulic separator to make domestic hot water for the building, combining thermal loads to one boiler to enhance duty cycle & improved net efficiency, but those options are a bigger redesign than just swapping out zone valves for circulator pumps with a fabbed-up hydraulic separator.