I have a Radicale server that uses CalDAV and iOS works fine with it.
Thank you for the advice to keep pushing... I eventually realized I was able to add an account in Apps>Calendar>Add Account>Other, link my purelymail account, then add the calendars it creates to my Thunderbird instance. Unfortunately it doesn't seem like I can keep my calendar, so I will probably just move stuff to the two calendars it created for events and reminders.