Draw knobs require both in and out motion. Generally a bi-directional solenoid mechanism would require 3 wires, a common, an up (out) solenoid and a down (in) solenoid. With only 2 wires, my guess would be a bi-directional operation would require a voltage polarity reversal which would complicate the setup altho no doubt could be done.
Do you already have draw knob heads and mechanisms? You would also obviously require a switch for each draw knob and a means to fasten and connect all this to a stop board. The basic idea appears feasible but requires quite some amount of mechanical assembly work. Commercial draw knob assemblies are quite durable and compact for good reason and their cost reflects that as well.
Since it appears that the solenoid is mounted at 90 degrees with respect to the mechanism, (versus in line) thought would be required as to how to orient the assembly based on proximity to other assemblies.
A minor consideration would be how much noise (klunk) is generated.
Rgds,
Ed