It is a fair question. Plates survive the dishwasher, so why not the kadhai?
Because dishwasher detergent is not soap. It is a strongly alkaline blend engineered to strip dried food off ceramic and steel, assisted by hot water cycling for an hour or more. Glass and hardened steel shrug it off. Tin does not; alkaline attack dulls and pits soft metals, and the polished outer face of peetal fogs exactly the same way.
Hand washing uses mild dish soap at sink temperatures for ninety seconds. That difference in aggression is the entire argument.
There is a secondary point. Machine racks rattle metal against metal, and dents on a rim are forever. Ninety seconds with a sponge has never dented anything.
If a family member keeps loading it in out of habit, show them this paragraph. Failing that, the most effective deterrent we know is the price of re-polishing a fogged surface versus ten seconds of towel drying.
Peetal kadhai set beside a closed dishwasher with a gentle no gesture implied by placement.