No. There is a thermostat for this purpose. You can open a fridge 0 or 50 times per day and it will keep the set temperature (on average). If things freeze in your fridge, it does not mean you don’t open it enough – it’s misadjusted or faulty. And the evaporator (cooling) tubes are in the back wall, on which water condenses and does get cold enough to freeze sometimes. Anyway, the droplets fall into a drain and a narrow tube carries them out, usually onto a little tray on top of the compressor, whose heat helps the water evaporate. Replacing the bulb with an LED has no effect besides decreasing energy usage.

In fact, if an E14 LED burns out, you can often remove the plastic cover, find the burnt LED chip (if that’s what failed; look for a black spot), pry off all its plastic parts and short the contacts by scraping the aluminum/copper pads together with a knife. A soldering iron works better but I cannot bother waiting for it to heat up. Another LED chip will likely burn out sooner or later if it runs hot again but it will not get over 50 °C in usual fridge operation. Check if it fits into the fridge and add an aluminum foil reflector to redirect light if it helps. I did this in some Liebherr, Zanussi and Electrolux fridges, which I believe are also available in neighboring Germany. I use 2~4W “candle” bulbs for this because they are tiny without the plastic diffuser and not great for general lighting in most fixtures. If you don’t have failed LED bulbs, there is a free selection at dm or Lidl near the entrance. Be quick and sneaky but don’t take any CFLs (center of mass or shining a phone flashlight through the milky glass helps reveal a CFL tube if coiled inside). If you accidentally take a CFL or fail to fix an LED, return it on your next visit with an obvious hand movement so that your activity appears legitimate overall.

Old incandescent bulbs actually used to melt snow for streetlight and this had to be addressed when switching to LEDs.

In a large fridge a small incandescent lightbulb won’t make much difference since it turns off when the door closes and would have a relatively small thermal mass, which I think is what you’re saying. That said I don’t think it’s accurate to say that “all of the heat is in the filament.” Heat spreads, and it will leave the light bulb. If you have ever touched an incandescent light bulb that is on you would have no doubts about this! Easy bake ovens used to use incandescent light bulbs to cook things, people leave their oven lights on to keep their oven warmer to let bread rise. Larger incandescent light bulbs can absolutely warm up a full sized room, 60W or 100W is a fair amount of heat. There are stories about extremely temperature controlled rooms where they would turn on a 100W bulb when a person leaves it because a person produces about 100W of heat.

Yeah, I can’t touch a 100w incadescent bulb that’s been on for an hour, but an equivalent LED bulb could be on for days and I still can touch it without issues.