Not as much overkill as the wrong tool for the job. A typical PID output is designed to go to an SSR and/or contactor to control the electric device, whether a heating element in an electric kettle or a compressor for a keezer. They also cost $40-50, not including SSR, heat sink, contactor, etc, which can total over $100, and require some extra time/labor. You can get an "aquarium controller" like an STC-1000 for $20-30 that was designed to maintain even temps with 2 110V loads (heating and cooling), and is very simple to wire up (tons of info on other threads here).
Of course, some aren't as blingy as a PID.