You need to understand the relationship between heat flow and temperature.
A system's equilibrium temperature is whatever value that occurs when the heat flowing in matches the heat flowing out, keeping in mind that both flow rates are in some way dependent on that temperature.
The "bare" Peltier can reach 10°C because heat is transferring to it slowly relative to how fast the active circuit can transfer it away (to the hot side). When you add the heatsink, heat can flow into the cold side from the heatsink itself — as well as from the ambient air as the heatsink cools down — much faster. This raises the equilibrium temperature.
In other words, this is just a fact of physics. If you want to have a higher heat flow at a given temperature, you need a bigger Peltier device (or more of them).