Sure, they have a say. That's why the Atherton home page has this text:
> The Town of Atherton desires, insofar as possible, to preserve its character as a scenic, rural, thickly-wooded, residential area, with abundant open space with streets designed primarily as scenic routes rather than for speed of travel.
Notice the "as possible?". Cities only exist because the states gave them a charter. Cities must work within state law. The relevant state law is the California Vehicle Code.
You requested: Define "public".
Public here means anyone. "Public to Atherton residents" is not public. Palo Alto has its own openspace area which is only open to Palo Alto residents but not to the public. "Public" even includes a visitor from another country who has never paid a penny in US taxes. Any other interpretation is incompatible with current legal use.
The CA VC says that local officials can close or divert traffic on a road, but only for certain circumstances. (Regulation of Highways, §21101 if you want to read it yourself.): Local authorities, for those highways under their jurisdiction, may adopt rules and regulations by ordinance or resolution on the following matters:
Reading the CA code, there are public roads and private roads. A private road may be open for public, in which case the CA Vehicle Code still applies. If a private road is truly a private road and outside of the CA Vehicle Code, then the city of Atherton will need to make a lot of work to prove that, and a lot more work to develop its own regulations for how to manage those roads. So it's safe to assume that all of the streets under discussion are "public streets" covered under the CA Vehicle Code.
The VC takes a dim view of restricting access to streets. There's only a few ways for local officials to designate something as "not a through street." The most likely is if "The street has an unsafe volume of traffic and a significant incidence of crime." and "The local authority makes a finding that closure of the street likely would result in a reduced rate of crime." That doesn't seem to be the case here.
By "dim view" I give as an example §21101.6.: Notwithstanding Section 21101, local authorities may not place gates or other selective devices on any street which deny or restrict the access of certain members of the public to the street, while permitting others unrestricted access to the street.
I think that helps show that "public" means "public to everyone", and not limited to certain members.
This "dim view" is even more so for arterial streets, which is now almost the only want to get through Atherton from Redwood City.
Earlier you wrote: "maybe they just don't want people using residential streets as highways?"
There can be a lot of things that they may not want. They may not want any streets at all. However, they have to work within the law. The law doesn't seem to allow a posting of "no through traffic."
If the traffic is dangerous because of excessive speed, then they can set up police to enforce the speed limits, or they can get the roads evaluated to see if the speed limit is too high. However, that does not appear to be the case here.
I think it's more useful to read what the law actually says than to try and guess what things might mean. For situations like this, the definitions have been worked out in far better details than I would ever have considered had I just stuck to debating about what "public" might mean.