My Calendar app entries are read-only on iPhone

Hmmm, that’s not good.

The calendar entries that are read-only are almost certainly housed in a calendar to which you don’t have write access permission.

Share a calendar on iCloud.com - Apple Support

Aside from that, it’s strange that you list both “me” and “iCloud”. One would expect a person’s calendar entries to either all be in iCloud or none be in iCloud.

Where might these entities be housed? Check Settings > Apps > Calendar > Calendar Accounts. Hopefully there is exactly one housing place listed there.