OpenStreetMap is just data, if you don’t want to rely on a third party you will have to keep a copy of the data yourself and create your own maps (what you intend to do anyway).
There are many options how to do it, depending also on the decision how to present the map (vector tiles or raster tiles). Regarding the privacy concerns, AFAIK if you activate location based services you will unavoidably share your location with either apple or google provided you are on iOS or Android, maybe they say it is just in “anonymous form to improve the service”, would have to analyze the terms of service for this. And you will share your approximate position with your mobile provider regardless of using location based services, as soon as you book into a cell.