Walkhighlands

This looks like a good easy walk to do to start our holiday in June.
My concern would be the
PRIVATE PROPERTY - Access For Residents sign at the gateway before you even get to the bridge.

Assume that's just for cars?

The sign - and many others like it on private roads - refers to vehicular access as mentioned in the route description which tells you where to park.

You can read this official website for details on your access rights in Scotland.
https://www.outdooraccess-scotland.scot/practical-guide-all/walking