I agree with you. When it was just a footway, presumably drivers using the private access were expected to give way to pedestrians, just as they should when using a footway crossover. I don’t see why the rules should change because you are adding cyclists to the mix.
And a practical point is that cyclists are not known for being particularly keen to observe regulatory measures. Would placing markings actually have any road safety benefit, or is it just to relieve the private access user of the need to be vigilant and absolve him of responsibility for any collision?
There is actually no obligation on authorities to provide give way markings even at junctions between public roads, so there is clearly no reason that you have to do so here.
Any other thoughts from anyone?