Markijbema's comment on the original question touched on one aspect that nobody has mentioned:
"A seat is easier to combine with a typical dutch bike, which is less race-y than the crazy bikes people in other countries seem to use for daily use"
The construction and design of the bicycle determines how well it handles a load, either over the rear wheel, front wheel, or distributed evenly (in this case a 20-40 pound child). I'd like to compare three bicycles of roughly the same price point and frame/wheel size (~560mm, all measurements in mm or degrees).
The bikes: A) Surly Long Haul Trucker--700c, heavy duty touring, B) GT Zaskar Sport--29", hardtail mountain bike, C) Specialized Allez Comp--700c, speedy road bike
((I couldn't find any geometry details on Dutch-style cruisers, but I feel that they will be more in line with the Long Haul Trucker.))
Chainstay length affects whether your heels hit your pannier bags and the size of tire you can ride. (A) 460, B) 445, C) 405)
The wheelbase affects handling. A longer wheelbase generally increases stability. (A) 1056, B) 1110, C) 986)
BB Drop determines BB Height as well as the length of the stays and down tube, affecting the overall flexibility of the bike, as short tubes are stiffer and more responsive. (A) 78, B) 65, C) 69)
Fork rake influences steering stability. Larger rake offset means more stable steering. (A) 45, B) 39, C) 45)
Head tube angle influences steering response in concert with fork rake. (A) 72, B) 72, C) 73.5)
Each bike has its virtues, but not all bikes are designed to carry a load.
While the Allez is a fast and nimble road bike, it would feel twitchy and harsh with a child in a seat. There is no provision for mounting such a rack to this bike and frame clearance allows for 28c tires maximum, increasing the likelihood of a flat (also, this wouldn't be a fun ride for the kid!)
Mountain bikes like the Zaskar are becoming common for general riding. They're reasonably comfortable, have quick handling, and allow for large cushy tires. Like the Allez, however, that quick steering response is not a good feature when hauling anything. Also like the Allez, there are no provisions for mounting racks. With a high bottom bracket and cushy tires, your seat will be much higher and, consequently, you won't be able to put a foot down without dismounting.
The sensible choice would be the Long Haul Trucker, but of course it was designed for just such a load. It accommodates cushy tires, fenders, lots of racks, and a good deal of weight. It isn't as nimble as the Allez and can't ride over fallen trees, but it is very good at moving things around.
When it comes to choosing between a trailer and a mounted seat for hauling your kids around, not all bikes are equal.