Since the northern road is estate traffic only, the restrictions are the right thing to do. The Google places for Haddo House and Haddo Chapel are closest to that segment. If the Waze place were deleted, or the Google links removed from it, or new Google results on the estate are added to Gmaps, then the absence of restrictions could lead to Waze users being directed along the estate-only road.
Admittedly this could lead to bad routing for genuine estate traffic. But these users are few and should know where they’re going anyway. Given the “satnav is wrong” signs, they plainly have an issue with this. With the restrictions on the segments, we can be sure that this does not include Waze. 
As for the segment type on the southern segment: as I explained, what type it is makes no difference to routing as long as it’s routeable (not walking trail, pedestrian). Whether it’s offroad, private, PLR, unpaved, narrow road, street or freeway - if it’s the only unrestricted segment that lead to the destination, it will be used.
The gate I referred to is here. Alright, I know there are only gateposts and no actual barrier, but I would interpret that as being clearly the entry to private land, along with the signs for Haddo House/Estate. Having it as private makes a small but noticeable difference in the app which I thing is appropriate. And, as I said, the fact you’ve changed it to street means it’s now a segment type that is expected to have a speed limit, but doesn’t and WME Magic will highlight that.
It’s also an un-named segment. At UK Meetups, Waze HQ have occasionally used a metric of un-named streets to measure the “maturity” of the UK map. This is mainly stupid, as it ignores the fact that vast numbers of UK rural roads genuinely have no name and I never set roads as private just to avoid that. But in this case, I think Private is a justifiable segment type and again removes a potential problem that WME Colour Highlights or WME Validator will pick up.
Let me put it another way. I don’t remember if I set the segment type to private - the history doesn’t say. It might have been private before I touched it. But if it was private before I came along, I would have seen no reason whatsoever to change that. 