A Honda Odyssey will tow up to 3,500 pounds "when properly equipped". My brother traded in his old Odyssey not long ago for a new one. And he used to have a smaller popup. Now he has a bigger popup.
He would tow the smaller one on "local" trips but for any longer trips, they use the F-150.
The Town and Country/Grand Caravan/Routan can tow 3,600 pounds. The Sienna, Sedona and Quest 3,500. The others don't appear to be rated for towing at all.
So I agree, no good choices unless you go to an ultralight and keep your overall weight down. We just tent camp with our Quest, but we don't have kids to try to keep dry.
There is however, the Nissan NV Passenger Van. That puppy has sliders and can tow 7,000 pounds, 9,500 with the V8. But you'll wind up carpooling the entire soccer team.
I'm a bit partial to Nissans but if you want to look into cargo vans, there's also the Chevy Express and Ford E Series and the Mercedes Sprinter.