If we're having salad, I hard-boil eggs in a small saucepan along with potatoes.
A glimpse of social history from 60 years ago may explain why I don't see any problem or use a special egg pan.
When I was a girl and lived on a farm, in the days of the Egg Marketing Board (way back in the middle of the 20th century), it was often my job to help 'do' the eggs in the evening. Our hens were locked up in hen-houses overnight and let out early every morning; they laid in straw-lined nest-boxes.Their eggs were collected by hand, twice a day, into buckets with a bit of straw in the bottom.
After the tea-time washing-up was done, the water was left in the sink and the egg buckets lifted on to the kitchen table. We aimed to have the job completed and all clearing-up done by the time The Archers was over (in those days it ran from 6:45 to 7pm).
We sorted the eggs by size, and put clean ones straight into the grey cardboard trays that held 30, or used a damp cloth to rub off eg mud and light soiling first. Very dirty eggs were washed in the cooled water left from washing-up, and sometimes even rubbed with Vim powder to remove residual staining, before being dried and put in the trays. Only cracked eggs, wind eggs, or those so stained that we couldn't make them look clean were kept for use at home. (I think my father gave the egg money to my mother.)