An all wood lathe for wood working, cool. Really neat to see old technology and it is so simple!
But is only cuts half of the time and there is no inertia. I'm picturing turning a manual kick pottery wheel on its side and using that for a lathe. Manual pottery wheels usually have a fair amount of inertia and that would make cutting smoother. Something like this.
Granted this is a heavier and more expensive option.
What you're describing is a little more like a treadle lathe.
If you were to put the pottery wheel below the drive shaft and use a belt I would guess that you would get a lot less vibration in the work piece.
Fairly compact treadle lathe
This guy has a great set of videos. If you haven't seen any, you should check them out.
The spring pole lathe is several hundred years old. Although it has the drawback of rotating in two directions and only cutting in one, it has the advantage of being extraordinarily simple to build and maintain. Use, well, it's a lathe, your mileage may vary
Treadle types, with or without flywheels, are more complex machines. They are an improvement, but require a different level of technology than the pole lathes.
Unless you have a LOT of flywheel mass, you still have to vary tool pressure with pedal strokes on a treadle lathe. Look at the horsepower ratings on motorized lathes and figure what fraction of that you can provide with each stroke while balancing on one foot and manipulating the tool. Sharp tools are essential but make it trickier.
I've done a bit of turning (electric powered, and the only powered benchtool I use now) and got a chance to use a springpole at a craft fair. I'd say the learning curve would be pretty steep to learn the basics of turning on a spring pole. On the other hand, you'd figure out a lot of efficiency tricks ... And failures (AKA it's kindling now ) would be less catastrophic.
A spring pole lathe is on my 'to do list'; pretty far down, unfortunately.
I'm looking forward to seeing your build and results.