Certainly, we should keep the common names. Common names are part of culture, and are interesting in their own right, in that they capture some aspect of the human relationship to the world of nature, often going back many centuries and reflecting a distant, bygone time. Thinking about medicinal plants “lungwort”, “bladderwort”, or names that reflect the similarity of a plant to others, with a modifier that distinguishes it, e.g. “false lily of the valley”, or a different habitat than species that seemed similar, such as “mountain mint”, “mountain beaver”, or smelliness “skunk cabbage”, or just by location.
However, I’m not in love enough with them to set them as the default to show on iNaturalist. I just appreciate seeing them alongside the scientific names, and I use them in comments when it feels more appropriate, as in the case of common, local, and familiar species.
This is likely to happen. Unfortunately this is a drawback of the use of common names. Common names do not have any taxonomic meaning, even though there is sometimes a correspondence between them and the morphology/ecology/behaviour/chorology of a given species. So, it’s likely that certain users that are not experts could misundertand the value of a common name or be misleaded by a peculiar common name.
Anyway, the same can happen with scientific names which are univocal way of tracking species independently of their meaning.
I think that this is one of the reasons why common names should not have a priority over scientific names while in iNat it is the exact opposite. After all, there would be nothing bad in fostering the use of the scientific names among users.