No, without operational refrigeration units to force radiant heat loss, there's not enough surface area to radiate the excess heat into space. The spacecraft had a cooling system that needed power, and was shut down. Therefore the heat would have built up. Maybe not enough to kill anyone, but they would have gotten hotter, not colder.
Vacuum is not an insulator. Heat transfer occurs by conduction, convection, and radiation. In a vacuum, conduction and convection do not occur. But radiation is maximized when there’s nothing to block it. Spacecraft don’t lose heat by conduction (because there’s nothing to conduct it), or by convection (which requires moving material in a gravity). Spacecraft experience maximum heat loss (or gain, if in sunlight) from radiant heat. Even on earth, deserts experience large heat swings from day to night, because a dry atmosphere doesn’t block radiation very well, so a desert is exchanging increased radiant heat with the sky. At night, the sky is at close to absolute zero, so the desert loses heat quickly.
That cooling system was designed to cool the electronics in normal operation, which generate heat. With those shut down, the cooling system is no longer necessary.
Heat energy, left alone, will always dissipate to its lowest state. Keeping a spacecraft at a certain temperature range is a balance between heat absorption, generation, and dissipation. Before the accident, the generated heat (electronics and human) and the absorbed heat on the sun side, were balanced by the cooling system and radiated heat dissipated on the cold space side. After the accident, the spacecraft generated heat was greatly reduced, the cooling system was off, and the new equilibrium temperature was uncomfortably colder.
If the scenario had been just cooling system failure with everything else still up and generating heat, your speculation about temperature increasing would likely be correct.