It's a lot more expensive, but measuring navigation distance rather than straight line distance would avoid the "river" issue. Although depending on the routing engine and dataset it might well introduce more issues where points can be really close on foot but the only known route is a driving route.
To share a related river issue, a few years back the city of Gothenburg, Sweden switched how they allocate students to a school. It used to be the student would get a school in their specific section of the city ("statsdel"). The new one lets you choose a school, specified in rank order, even if it's in another part of the city.
You could select from a list of schools, with distances given as straight-line distance (or perhaps as route distance? I can't tell from the articles I've read), which meant some of the schools across the river were considered "close".
In one case, a student had a 45 minute commute to get to school, due to waiting for the ferry. The parents listed it as their 5th choice, based on the stated distance. The more critical factor should be travel time, but the computer system doesn't take mass transit schedules into account.
As a contributing factor, the list of schools did not include the name of the stadsdel in the list of schools. This extra sort of reverse geocoding - trivially available - might have helped them realize the issue.
