Lightning is just way too powerful. Electro-magnets (and motors and generators) are made of many windings of fine wires that the lightning just vaporizes.
Lightning already has it's own EMP effect, that's how lightning detectors work.
Since lightning rods are able to withstand multiple strikes over many years, and they're generally just made of copper or aluminum, I bet dollars to donuts that 2000 mcm copper cable would be fine. Nearly 50% increased diameter compared to normal lightning rods means a much greater ampacity rating, especially when you consider that it's a stranded conductor. Though you do have a point in that shaping it into a coil to get the magnetic effect does introduce substantial inductance that would have to be dissipated by heat (hence the larger wire size). Surely 1 loop would not be obliterated by a single lightning strike, but how much magnetic force would that produce? 1,000,000 wraps would definitely be too much for the cable to handle. Some middle ground would have to be determined. The power generation would be on an isolated separate electrical system so its windings would not be exposed to the lightning blast, only the magnet-o-piston.
Some unknowns need to be solved in order to determine basic schematics for a prototype or an estimate for how much weight could be lifted magnetically. A minimum low ball estimate is that using extreme caution, a system could propel a 100 kilogram mass 1 meter into the air from a single lightning strike. That would be about 1000 KJ of potential energy gained, which is only a fraction of a kilowatt-hour, but is still a net output of energy with nature providing all the input once the system is in place, and proves the concept.
With some rough napkin math and google, the average lightning strike produces about the equivalent of 277 kW-h of electricty, which is enough power to run my electric stove, PC, lights, hot water, and fans - for an average autumn month in a temperate biome. Even considering only a 1% energy conversion efficiency, given some storms strike 20+ times a minute and can last for hours, I hold my position that this is a valid idea ripe for invention. Construction may be prohibitively dangerous for humans, but some EMP shielded construction robots should be fine. I suppose the world is waiting on those to be invented first. Perhaps starting construction with the power generation system, then the weighted magnet piston, then the coil, before finally attaching the vertical attractor rod would be the safest build order? Built in a lightning hot spot would give the most return on investment. However the likelihood of lightning construction casualties has potential to negate any positive effect. Someone needs to get to work on EMP shielded construction robots. Or maybe we just need to understand lightning better in order to know how to operate more safely in close proximity to it. After-all, people safely work in the empire state building's top floors despite 23 lightning strikes to the building each year.