Should I give my router a “break” between passes?

Short answer: yes, you should do breaks.

Longer answer: it depends on how how does a motor and a bit gets.

There's a nominal temperature for a motor depending in its class - the highest temperature a motor can be indefinitely at without any degradation in its primary functions. You could look it up for a specific motor, I picked mine to be ambient temperature +60˚C, so if my workshop is at 25˚C, I would let a motor warm up to 85˚C. Every 10˚C over this value will reduce motor lifetime 2x. Some machines (e.g. planers) have overheating protection, my router doesn't, so I cap the highest temp at 95˚C

When your router idles (that is rotating without load) it takes very little current and pretty much is not heating, so I would switch it off only if preparing next pass takes more than a couple of minutes or makes me turn away from a router. Switching a router on and off repeatedly will heat it more than idling, also it's not good for bearings.

This being said, a bit would get way hotter in a way shorter time, it fact it's totally possible to burn a bit in just a few moments after a cold start. So if you are doing raised panel for a whole kitchen in one run - I would do breaks after 4-5 passes - swipe shavings and dust, move ready pieces away, etc.

The motor/bearings in the router aren't generally much of a concern for overheating. However, router bits can definitely overheat. Particularly, if you're using a high speed steel (HSS) bit overheating it will cause it to loose hardness and dull very quickly.

Carbide is generally able to dissipate heat better than HSS, but if you get them too hot they will be more prone to chipping.

I normally always switch off all my power tools in between setup and next pass especially when working with large pieces. The power switch is located and convenient to access. So turning off the tool is more for safety. All prep work would mean the power tool is turned off.

With a router underneath the router table, without a separate convenient switch location, this may not be easily done.

I am building a router table now, that will have an easy-to-reach push cutoff switch.

To answer your specific question about cooling, you should not be feeding the router that hard that it requires a cooling-off period. If you see that the router is starting to heat up, then a cooling off period is needed. Your router may cool off faster with the router on with the fan blowing air through the motor.