Managing a Child's iPad Without Another Apple Device?
What’s wrong with setting a parental restriction passcode and not letting your child know the iCloud password for the account you set up?


https://support.apple.com/en-us/HT201304


Pretty much everything you have set up is easily achievable in solo mode. You could also sign up for an MDM like Jamf Now 


https://www.jamf.com/products/jamf-now/


That adds far more options than a family AppleID and has many classroom features and integrations. It’s free for op your first three managed devices and Apple specific MDM. 

Longer term, you might pick up a used phone or iPod touch that can run the latest iOS and manage the shared family AppleID but many of the family features work best with an iOS device for you to manage things remotely from the iPad you leave with your child. 

You have discovered all the drawbacks so your problem statements leads me to believe you don’t have any more major surprises. Two last recommendations, if your child is less than 13 you need to manage their password and agree to all terms for them. Use your account and birthdate and just don’t give them the password ever. Use a password service like 1Password to save all details about your AppleID, the birth date you entered (people often mis type that when they are wanting to get their account set up), security questions and answers and the password, passcode, unlock PIN, parental PIN, etc...  document everything and keep your proof of purchase if you need Apple to unlock the device if iCloud gets locked by your child entering the password wrong too many times.