You have to get the water OFF the road surface ASAP. Given the slope of your road, you also have to think about the uphill side hill, which, given you've been removing its downhill support, will sag and erode at a HUGELY increased rate. IMO, I am afraid that you bit off more than your little tractor will be able to adequately do and maintain. At a minimum, put in diversion berms along the road surface to keep the stream from becoming a river.
Without an uphill ditch, you will washout your road at every low spot along the hill. Given the steepness of the road, you will create a 'downhill stream' that will either break off the shoulder, or finally runout at the end of each straight section, thereby washing out your switchbacks. If you are really wanting a road, I think you will likely need several culverts along each stretch and at each switchback. I also think you will need a very good gravel base which is angled inward, towards your uphill ditch. If you crown it or angle it to sheet off the road towards the downhill side, you will washout your shoulder with each rain.
This is a real project... and I am glad you won't be living at the top.
Good luck and take pictures.