I think you're having trouble with it only because of how you're trying to look at it.
Don't try to imagine "spreading force" over time. Just think about how long the force lasts !
Like . . .
If you push your doll stroller for 5 seconds, you'll get it up to a certain speed.
Next time, if you push it again with the same force, but you keep pushing for TEN seconds, you'll get it up to a faster speed.
Technically:
(force • time) = impulse.
You gave it more impulse the second time, so it got more momentum from you.