That's a crane. Cranes are low constrained
What you are proposing is a portal crane. Have you ever seen a container portal crane in operation?
Containers are strung up at the 4 corners and lifted from long cables. They move on their path because the container is super heavy and they have had years to finetune the acceleration profile. But the container sways and swings from side to side during airtime because the cables hang!
Only in a container lifter, where the container is carried by a gantry that is constrained on 4 corners and very heavy to keep the cables tight and taunt at every moment, you get the needed constrainment, and now we are talking that the claw lifting the container weighs in the order of tons.
Scaled-down to a printer, a napkin estimation of the gantry tells me that it would need to weigh in the order of about half a dozen kilos to ensure that nothing in the printer path could force the nozzle up - and the motors you need now are actually high power servos as you find them in heavy CNC equipment. However, that's for a conventional crane: the printhead hangs straight down, the mounting point of the crane is moving, and the high mass is to try and reduce swaying as we go really really slow.
extreme example of low constrainment: Hangprinter
The same problem was present in the hangprinter - in the idea that is a free-floating gantry with little to no limitations. The main problem it faces is layer accuracy because of the length of the ropes. The suspension stretches as a factor of the length of them, making movement complicated, but solvable. By using polyester that has no real stretching and mounting the wires in a triangular fashion by three cables, the printer becomes self-stabilizing. But the free-floating makes it extremely vulnerable to imparted errors from outside, such as wind or just the building swaying. Also, its print volume is a cone.
Constraining the gantry
Of course, we don't want to have a free-hanging gantry like a hangprinter. But now we need to constrain the gantry. With a quad of rods, that'd be enough, similar to how they are in the haulers on the ground in container harbors. That constrainment also reduces the needed weight to make it resilient to shifting, but the gantry still needs to be somewhat heavy to prevent the shifting weight to alter the balance - the gantry itself should be at least three times as heavy as the printhead to minimize the lifting cables stretching differently when the printhead moves left and right.
However, the setup still suffers from some large downsides:
The smallest problem is the elongation of cables under stress. While this is dependant on the length of cable, so the lifting profile isn't exactly linear, the right material choice can fix this.
A bigger problem is the diameter of the spool changing with the number of windings that the cable is coiled up over, adding more non-linearity, and this is hard to predict in the worst case. If one side gets its cable into the next layer while the other doesn't, this could force a tilting force onto the gantry.
Tilting forces of enough strength could, in the worst case, press the bearings in such a way against the rods that the result in a lock-up situation where the gantry isn't going anywhere. This could be solved with some engineering ingenuity and proper tolerancing.
Another problem is, that while constrained to the down direction, the setup isn't constrained to the up direction, making it possible to push the gantry up to the top, but the weight will get it back to the print level again. However, such a bump might result in the force on the motor losing a few steps, giving us layer shifting in the vertical. This is an outside-induced error though.
To mitigate most, if not all of these problems, you would need to use cables in both directions (one to hoist, one to prevent lift), which basically makes this a continuous belt-attached gantry and mitigates any requirement for a heavy gantry to keep the cables stretched. A well-designed set of loops also would prevent tilt-locking the gantry as the cable can't get mis-wound - possibly even all driven by one motor.
Mechanically linked setups
What you are looking for are mechanically linked/synchronized Z-axis'. That actually can be done without needing pulleys. A single motor can turn both leadscrews easily if it was connected to both with a proper motion train. Examples are using a chain and gears, or just a gear train using a horizontal axle and two pairs of angled gears to turn the leadscrews. Take a look into how heavy machinery synchronizes axis'.
If you want to go with a pair of belt-loops, you could have a single motor turn an axle upon which the gears for both belts are mounted, resulting in synchronized movement of both belts and the required constrainment on both belts. An example using 4 belts is the machine mentioned by 0scar. However, keep in mind that you want non-stretchy belts for this and to keep them tuned well.
