You figured it out, short trips will cause major damage to the engine sooner or later, and chances are it's gonna be a lot sooner than you'd expect it to be
Besides my XT600 which I use for longer trips and journeys, I own a Honda XL185S for everyday business, the city and real dirt. It's 25 years old and was dirt cheap. Incredible mileage as well, 3-3.5l/100km is what it usually takes and that's half of what my Tenere needs in the city.
To sum it up: If you're looking at something new how 'bout a 200-400cc bike - cheap, reliable and... well, cheap. If you're looking for something older same advice, look for 200-400cc
"Life is not a journey to the grave with the intention of arriving safely in one pretty and well preserved piece, but to skid across the line broadside, thoroughly used up, worn out, leaking oil, shouting GERONIMO!"