Replace Furnace or Stick with Dual Zone Setup?
It's not clear to me how the second zone turns on and off.
Normally I would expect it to work by controlling the air flow.  So one zone (the one that gets chronically cold) is heated whenever the furnace comes on.  The other is heated when a damper to that zone is opened.
The problem with this:
Forced air furnaces are designed to output a certain volume of air.  Restricting the air flow results in a higher output air temp and a loss of efficiency.  In addition metal vent grates can be painful to touch. (But do a nice job of drying wet mittens.)
So a better way would be to keep the air flow constant, but split the flow between the two zones.  Visualize a 1x2 foot duct that splits into two adjacent parallel 1x1 ducts, and has a 1 foot square plate that can slide to any position from covering all of duct 1 to all of duct 2.  I've seen devices like this in commercial HVAC, but not in residential systems.
If you are doing a major refit to add AC anyway, you may want to take advantage of future differential prices of electricity:

Use the AC hardware at night when power is cheaper to chill a large tank of water to near freezing.
Circulate that chilled water though a fan coil to provide cold air for the house.

In winter the AC is turned off, and a small high efficiency gas water heater is used to heat the tank of water.  Since the tank has no circuity or fittings, it has very long life and is fairly cheap.  Insulate it well.
Potentially this also allows you to add warmer zones to the house. E.g. hot water under the bathroom floors. Hot water towel warmers. (Note that the requirements for embedded hot water, and radiant hot water take much different temperatures.) This however makes the system more complex.