It’s actually really good. I just wish the maps were as up to date as the big three (Waze, Apple, google). It would be awesome if each city, state was required to publish map data to openstreetmaps. I know my city has everything in arcgis and you can browse it. But they should definitely publish to openstreetmaps.
I have found OSM data (and therefore Organic Maps) more up to date than the others, at least for roads. In the past six months for me that was three different cities, and two different countries. Small sample size, but including my city there were several places that google had not updated for years.
As for your city to OSM integration: OSM does not take data dumps. But your city could encourage people to update the data in OSM or better still leverage OSM as a data truth and curate it, and load ArcGIS datastores from it.
You can help using Street Complete which is a very easy to use android app that fills in the details of whereever you are at: Is this road paved? What are the business hours here? Is there street lighting? etc.
The things is you can't just "publish to OpenStreetMap". OSM is an integrated dataset, and he integration happens by actual humans. Bulk operations are possible, but hard to get right and usually not actually massive. So you need a decent volunteer base to get the data good enough. Some apps do merge OSM data with official data before showing it to the user, for example to offer full building or address coverage.