Imagico wrote a blog post, Rethinking road access restrictions rendering | Imagico.de - Its a complicated matter, packing it all in a picture.
I’d hesitate though, to drop it: It is one argument when landowners/managers complain about forbidden paths - One can point out and say: Look here, the ban is shown on the map! (If you want the path to disappear, just plow the area 
)
It isn’t even car centric, as most of the restrictions actually apply only to vehicle=* (which the iD editor doesn’t even shown as option in it’s form) and not access
Reading some the comments, it seems the reverse might be clearer. So: access=yes foot=no bicycle=no This assumes the traffic is by default motorized and that any pedestrian and non-motorized bicycle traffic is prohibited.
I guess for now I’ll tag foot=no + bicycle=no but also tag the traffic_sign=* so in case future tagging schemes find these ways in case a superior tagging method comes around.
But how about on horse back or via non-mechanically propelled vehicles (e.g. horse and carriage)? Both seem prohibited in the OP but would be allowed in your tagging suggestion.
I think at a minimum, you’d need:
foot=no
bicycle=no
horse=no
vehicle=no
motor_vehicle=yes
To the more extreme, I have seen some access tagging in the UK for things like ski=no and snowmobile=no on regular footpaths which would never be thought even remotely suitable for skiing. Technically not wrong, but totally unnecessary.
in general access=yes is very rarely useful on roads (tag foot=no bicycle=no horse=no - though maybe one more negative is missing, that is why access=no motor_vehicle=yes has some benefits)
I find it quite useful for highway=service, which is otherwise ambiguous as to whether access is permitted. But otherwise yeah.
horse=yes in this context was a minor meme in the early days of OSM.
(“All taxonomies are folksonomies. I ain’t never seen a horse using JOSM.” – Louis Armstrong, possibly)
From what I remember, most of the interstate highways in the US say no pedestrians & no bikes, but nothing explicit about horses, horse & carriage, snowmobiles, etc. Sometimes common sense has to play a part.
Actually some do. It would be a pretty strange to see someone trying to ride a horse next to high speed traffic. I would imagine that the horse would be the first to protest.
Every state has its own version of this sign listing different modes of transportation, because state laws regarding highway access vary subtly by state. Some of the more rural states, like Kentucky, are explicit about prohibiting animals on foot.