You're just not going to get the same texture with anything that doesn't have spinning knives (blender, food processor, etc.) You might be able to get close with enough cooking + a masher, then a hand mixer (or do the old hand-crank ones still count as mechanical?).
As things go for the type of cooking you're dealing with, consider a stick blender -- they're much cheaper than a food processor, don't have the issue with leaking seals as a bar blender, and they don't take up as much storage space.
They can't quite do the same things as the others (bar blenders are good for breaking down ice cubes for smoothies that a stick blenerblender might choke on, and you can leave them alone while they work; food processors can shread or slice with the right disks, kneed dough with the plastic blade, or break down solid objects with pulsing to keep it from getting to pureed), but I don't think the others work for making whipped cream, either.