This is a nice lathe, and it would be a shame to do nothing with it.
I have a Hardinge lathe that i am updating, but i will be also changing the servos from brushed to AC servos.
the initial problem you will have is the interface to the drivers.
A lot of the old servos used +/-10V and this will be a problem with most of the PC based systems.
Then you have the PLC functionality that you will need for the tool changing turret.
This will require I/O, and even without the turret the lathe will need quite a bit of I/O, so will need additional hardware for the I/O and a dedicated CNC controller has PLC and a lot of I/O built in.
you also have all the Estop, collet opener coolant and other switches and pendant control already built into the panel.
With the PC based system, i would need a faster system than the parallel port, so looked at the smooth stepper, but was put off from the feedback it has had, i would also need some sort of modbus I/O expansion to handle the additional I/O.
Also with Mach 3 it would be open loop, so the controller would not know if the axis was out of position.

When you take all of this into account the costs between a PC based system and this dedicated CNC controller are not that far from each other.

I know someone who converted a hardinge lathe the same as mine to use Mach3, but it was not that straight forward, but can be done.
I considered Mach3 for the update, but then decided to go the with a CNC controller.
The controller that you pointed out would work, but you would most likely need to update the drivers because it only gives out clock and direction.
There are others that can use +/-10V but they are a lot more expensive, so it would most likely be better to change the motors and drivers.