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The issue with it not landing on the target temp is probably endemic so to speak. As your error (Temperature difference actual to target) reduces the drive to produce the correction also reduces and simple circuits like that often run out of puff a little way off the target.
Another design that can be very effective is the PID controller, often used in cars for closed loop control of idle speed or boost pressure in turbo cars. PID being Proportional, Integral, Derivative.
Proportional produces a correction in direct proportion to the error (Same as your OpAmp) and it will do the same thing and run out of puff a little way off the target.
Integral integrates the error over time, basically allowing a small error to build up a larger correction and drive the output harder.
Derivative produces a counter correction when the error is changing, basically limits the rate of change.
All three outputs are combined to produce a single correction figure.
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