I admit to not having watched the video, being more of the written word kind of guy (can you tell? 
) but my assumption here has been that your balancing spring can be tuned with a spring to reduce the load on the stepper. To avoid skipped steps you’ll still need to handle acceleration, even with the load balanced. This requires motion planning. Do you know about motion planning for stepper motors? When they talk about a fixed pause between halves of the cycle, that sounds like they aren’t using microstepping and aren’t considering acceleration.
This link explains that difference.
A stepper motor gives lets you control how many steps to take, and reduces or eliminates the need for position feedback. You often still want a “homing” operation where you can sense when the mechanism reaches a limit, but after that as long as you keep power flowing to the motor to hold it in its currently-desired position and it correctly takes every position you command, you don’t need to measure position.
That’s not really how it works. Modern stepper controllers subdivide the steps and give very smooth control through the steps, and can even adjust current many times per step to provide smooth motion. You would want to use these. The Trinamic controllers work this way, and they are what you would want to use for this.
A DC motor controller needs continuous position information, and whenever the mechanism senses that the position is wrong, it drives the motor toward the desired position. Corresponding to the motion planning for the stepper motor, the DC motor controller will have to implement a PID controller algorithm.
Here’s a video on PID controller theory that makes it intuitive based on seeing the impact on physical motion.
A DC motor will have much more torque for its physical size, basically because magnetic fields have inertia and a DC motor doesn’t have to change the magnetic fields as fast as a stepper. Also the DC motor has fewer poles than a stepper.
If you have a dead stepper, disassembling it will be educational! 
Good find on the taps, I was searching for t5 instead of tr5, oops!
The thicker the nut, the more bearing surface and the more friction. So it’s a balance.
You would want an undersized drill bit and a 7mm reamer, in a drill press or mill. You can’t really freehand drill with a reamer. Then you use an arbor press to insert them. Alternatively, yeah you can use a 7mm drill and just use some loctite and make sure the loctite doesn’t get into the bearing! I’ve done that (successfully!)
I assumed that the axle would be supported at every finger; 3mm would be fine with less than 8mm unsupported, but certainly not across the whole thing. But yes there are many many more bearing sizes, and you can choose what fits axle and finger.
A standard M4 is 0.7mm pitch. Other pitches are typically expressed with a qualifier, fine thread M4 is M4x0.5, and there are sometimes other custom pitches.
Usually “lead screw” should mean trapezoidal threadform.
You’ll see TR4.76 listed some places, but that is actually 3/16" ACME lead screw expressed in metric as far as I know.
I see NEMA8 (0.8" / 20mm) steppers with integrated TR4.76 lead screws on aliexpress. For example https://www.aliexpress.com/item/32622031618.html with a POM (delrin) nut.
OK wow I just went down the aliexpress rabbit hole. Here’s a linear actuator including a 10mm stepper and a 2mm lead screw:
https://www.aliexpress.com/item/4001154700714.html
18° per step; it’s a 10 step per rotation motor. I don’t see I can’t tell whether it’s 0.23, 0.25, or 0.5mm lead, and there are no current or torque specifications, but it’s a start.
Here’s one with better specifications and with a hex nut that would be easy to embed in a block. 0.5mm pitch, 25mm travel, 0.5A current, even has torque specified. Don’t worry about 5V — that says you’ll get 0.5A if you apply 5V, but normally you just limit the current and put a higher voltage across; driving it with 24V and 0.5A current limitation would be fine.
https://www.aliexpress.com/item/1005001348928816.html
Same motor specs (I think), 3mm x 1.2mm screw 50mm long, round nut:
https://www.aliexpress.com/item/4000383679456.html
If I were doing this, I would experiment with one of those in the one-string test bed and see whether it would work… If you could stagger the motors you could meet your 8mm pitch requirement, it seems. I’m sure that your balancing spring idea would be key to using such a small actuator! But it might be the key to success here.
Just want to make sure you know that while sketchup is really easy to work with, it’s also modeling surfaces not solids, and therefore is easy to specify parts that are “non-manifold” and can’t turn into models for printing or CNC. Since you’ve done 3D printing you’ve probably run into this already?
Also every conversion I’ve dealt with coming from sketchup has had imprecise measurements and I’ve had to rebuild it in CAD to have precise and correct measurements.
So yes, wire stretches. Heck, when you tighten a bolt, the bolt stretches! But the force on this cable would be much different than a guitar string; the cable is thicker and the force is orders of magnitude lower. You are not going to exceed the elastic limit for the cable, so it won’t stretch meaningfully, I think. And if the cables don’t have much bend, you’ll have very little friction in them.
No, you don’t want to push. Your mental model here is bike brakes not throttle cable. But it takes extremely little return force; I’m surprised how little force every time I play with a return spring. I just replaced a bowden release wire on a piece of furniture and the return spring felt like it was too small to do the job, but no actually it did the job just fine. I wouldn’t worry at all about that. It will be tremendously less force than what the motor would provide. You’ll just tune your balancing spring (you can tension it with a screw, like on a 3D printer extruder) so that you get good return behavior.