I lived in Heidelberg, with a population density of 1500/km², so about 4x the density of your place. There was a robust bus system, tram system, commuter train system, and then of course Germany's regional and intercity train systems. There were also plenty of public rental bikes and bike lanes. I could go anywhere in or around the city quite easily and quickly, as well as any other city in Germany (or the EU, for that matter). Trams had a frequency of about 10-15 minutes, rapid buses about the same, the bus stops by my house had a frequency of 20 minutes. There were suburbs up the river which also had phenomenal bus and commuter train access directly to the city and elsewhere.

The American town I live in now has a density of 900/km² and about ⅓ the population of Heidelberg. We just got our first bus last year and it runs in a loop once per hour. The train station was demolished decades ago.

I also lived in Sejong, with a population density of about 750/km², so about 2x your place. In addition to dedicated bike lanes on every major road and very large sidewalks, there was a extensive bus system and a very efficient rapid bus loop system as well. The rapid buses had a frequency of about 10 minutes and could take me to the other side of the city in about 15 minutes. The smaller buses also had a high frequency of about 15-20 minutes, depending on the bus. The train station in sejong is still under construction but it was a ~30 min rapid bus line ride to either of two train stations in neighboring cities to take me anywhere in South Korea.

Some of the other Korean cities with densities somewhat higher than Sejong, like Daejon which is about 2700/km², have really incredible subway/metro systems too.

In Germany, the nearby cities of Stuttgart (3000/km²) and Frankfurt (3100/km²) also had great subway systems, in addition to the buses, trams, bike lanes, and commuter trains.

The commuter and regional trains serve also the purpose of connecting much smaller towns and villages, which are far less dense but still served by good bus systems and such.

I do agree that America has sprawled so much as to make the transition more difficult. But great density-appropriate public transit is possible at low density.