The idea is generally to ensure the patient leaves the wound submerged for long enough. 10 minutes is absolutely enough for super minor burns. The problem is people usually don't understand the severity of a burn because the worse a burn is, the less it hurts. Fourth degree burns (as in charred bones) don't even hurt.
So for 1st / 2nd degree burns, it's safer to just give the generalized "30 minutes minimum."
As for a burn continuing to worsen, you DO have residual heat from the burn, you can usually feel it, like you'll literally radiate heat, unless it's a severe burn.
An untreated burn can absolutely worsen over time. The 'continues to cook' is a simplified way of saying that deeper tissue can be damaged from an untreated lesser burn.