It's also more amps than USB-C is rated for, so depending on the model of charger and fridge, this might not work reliably.
There are refrigerated coolers out there, which have essentially the same structure in miniature.
You still have cheapo compressors, insufficient or bad foam (some foams embrittle when exposed to sealed car temperatures, I don't know if that affects the R value but I wouldn't be surprised), but at least you can fix the seal, and have a form factor people are still familiar with.
It's only rated at 2.8A (34W) so it's within USB-C spec. It can probably exceed the spec when the compressor starts, but it hasn't been a problem.
Mine does have a thermostat control, and it can keep things frozen well. Used it with a portable 220Wh Li-ion battery to keep ice frozen during a party, it kept it frozen solid on a hot day in direct sunlight for 5 hours
As an engineering estimate the system wide COP of a peltier device is at most 0.05 vs pumped freon running around ten (to one sig fig). So with a solid state peltier device the 30 watt device sucks 1.5 watts of heat out of the fridge and dumps 31.5 watts into the heatsink (your room, eventually). A pumped freon unit like the one in your kitchen would suck 300 watts of heat out of the fridge and dump 330 watts into your room.
Anyway the definition of a BTU can't be more accurate than 1% for a bunch of boring thermodynamic reasons, but a good engineering estimate is one BTU is 1/3 of a watt-hour. So running your cooler will suck about 5 or so BTU out of the fridge every hour.
A BTU unit is very convenient as it changes the temp of one pound of water by one degree. And to one sig fig a 20 oz soda bottle is about a pound. So one bottle in an otherwise empty cooler should drop about 5 degrees per hour in your cooler. Figure a "few" bottles means they'll drop about one degree per hour. Yeah its gonna take "a couple hours" as you state to cool it down.
A pumped freon fridge could cool your soda close to 200 times faster than your cooler. That seems "about right" as a case of beer takes a couple hours to be drinkable.
I'm not sure if there are any residential "convection fridges" on the market. I'm sure the moving parts fan would increase future sales figures. Another interesting idea would be something like "NEST" thermostat to control the cooler temp while using as little power as possible. Might make some interesting startup ideas.
Anyway if you want to check your math, I did a quick experiment. My fridge cooled a 1.5 liter bottle of room temperature (25.6 C) tap water to 18.4 C in one hour, and 11.9 C in 2.
The wall to 12V brick I had was massive, and caused lots of noise on the mains (that could be heard in the speakers).
Eventually I found on Amazon a USB-C to cigarette lighter outlet cable, so now instead I'm using a compact GaN USB-C charger to power it, it works great!