There are plenty of other tile servers you can choose from, free or paid. Most provide a static tile service/API, usually called Static Tiles or just Tile Requests (if no vector tiles are supported).
A good catalogue of servers (usually called Providers elsewhere) can be found at the websites below:
Google Maps does not document a static raster tile server. Therefore, flutter_map is unable to show Google Maps.
There is an undocumented endpoint available, however it violates the Google Maps Platform ToS.
Switch2OSM also provides detailed instructions on how to serve your own tiles: this can be surprisingly economical and enjoyable if you don't mind a few hours in a Linux console.
However, this will require a very high-spec computer, especially for larger areas, and hosting this might be more complicated than it's worth. It's very difficult to fully understand the technologies involved.