I'm not familiar with the public notification app, or how it works. As you say, it's ignoring the filter, which leads me to believe that when the app is selecting features within a given radius is accessing the service endpoint directly and bypassing the filter.
A filter defined in the web map is stored in the web map, or alternatively can be stored on the layer item if you save the layer from the web map. As long as client apps are accessing through the layer item, they should apply the filter. It is the responsibility of the client app to respect the filter. However, the actual data source of the layer still contains the filtered data and it is accessible.
Here's something you might try and maybe it will work. Instead of applying the filter in the web map, create a view layer of your original layer and set a view definition instead. This actually enforces the filter at the data source level, as if the data isn't there.
To create a view layer, go to your layer item. There's a button to 'Create View Layer'
Create hosted feature layer views—ArcGIS Online Help | ArcGIS
On the new layer, go to the Visualize tab and in the left panel toolbar click the More Options button (3 dots). You can set a view definition here, which restricts features at the service endpoint.
Set hosted feature layer view definition—ArcGIS Online Help | ArcGIS
Perhaps if you use this layer, it will work as you expect.
Mike