An alternate intuition pump, at least if you're old enough to remember incandescent bulbs: consider how bright a 1000-watt bulb is, compared to how bright (in the visible spectrum) the 2-3000 watt oven element gets.
Look up the emission spectrum for a body at the temperature of a typical oven element: yes, it starts emitting some light that is visible, but the bulk of the energy is still in the IR band.
An alternate intuition pump, at least if you're old enough to remember incandescent bulbs: consider how bright a 1000-watt bulb is, compared to how bright (in the visible spectrum) the 2-3000 watt oven element gets.
Look up the emission spectrum for a body at the temperature of a typical oven element: yes, it starts emitting some light that is visible, but the bulk of the energy is still in the IR band.