This worm gear box is 3D printed ABS and uses a 40:1 worm gear set purchased via aliexpress for $20 shipped to the US. The rest of the parts, including a couple F608zz or R2280HH bearings, an 8mm shaft, and a NEMA-17 motor will cost about $20 if you don't already have them in your junk box.
The $40 cost for the drive unit compares favorably with the $140 OnDrives Rino 30:1 worm gear drive used in UMMD.
A complete Z axis will require the worm drive unit, 2x KP08 pillow blocks (@$2.50 each), 2x 40 tooth GT2 drive pulleys (@$4 each), and a pair of flanged pulleys, conveniently made from 4x F608zz bearings (@$2 each), and some steel or glass core GT2 2mm pitch belt (about $5/m), guide rails and bearings, and miscellaneous hardware.
Advantages of a worm gear driven belt-lifted Z axis include no bed drop when motor power is cut, high resolution, ability to lift heavy loads with a small motor, no backlash in the motion, no Z wobble.
The concept is sound. You just need to figure out where the execution went wrong.