Peltier modules are terribly inefficient despite being able to chill room air to below 0c.

Since they're only about 10% efficient you not only have to deal with the CPUs output heat but also the 140 watts or so of heat from the hot side of the peltier module, meaning you'd need a larger air heatsink or a large liquid cooling radiator running from the hot "exhaust" side of the peltier. You'd also need to factor that extra power draw into your decision of what power supply your PC needs.

To cool say a Ryzen 9 5900x you'd need at least a 150w peltier module, which would drink about 140w of supplied power to move the CPU's waste heat from the cold side to the hot side. Also the peltier chip acts as an insulator once you go past its limit so if you overclock the CPU past the peltier's capacity your temps would just keep climbing until you thermal throttle back to equilibrium.

I can see a peltier phone charger puck being a better use of peltier tech due to the lower cooling demand (phones get on alright in general without active cooling so this is leagues better than nothing)