Totally. You also have business offering an API to OSM data that they mirror and host themselves.

The limitation is for the openstreetmap.org website, that is not dimensioned to be heavily used.

> Would it be possible to use a local copy of OpenStreetMap database, not paying for API calls?

Yes, plenty of people do that. Be careful "OpenStreetMap API" is only for editing. There is other software in the OSM ecosystem which has it's own API (e.g. graphhopper has an API for routing). But sometimes people get confused and try to use the "OSM [editing] API" for all sorts of things.