I reworked my question to make it more coherent. This is Gen 9 where you use picnics for breeding, so there is no message like that.
With picnics, if you have multiple Pokemon in the party that can breed with each other, you can get eggs from multiple pairs. With my party, in theory, all Pokemon should have only been able to breed with Ditto. However, I got no eggs when both Meowth held everstones, even though both Meowth are female and Tandemaus is genderless.
You get eggs in batches of 1 through 10 when you check one of the baskets in the picnic. Egg Power 3 ensured I would get eggs quickly (as I did with my last attempt), but with my first 2 attempts at collecting eggs, I went several minutes without getting eggs, even when checking the basket periodically. So I can confidentiality say both Meowth holding the Everstone caused Ditto to not breed with anything.