Add the ability to turn on an referral-based registration from the admin panel and be able to set/manage a list of codes from there that could be used for new users when creating a new account. So when turned on, on registration page would be asked as required to type down a referral code to be able to create a new account.

Hi.

So, this would be a system where you can be invited by someone else and this person sends you a code and you have to fill it when you register, isn't it?

Well, you can do already something similar, but it would be a manual task.

You can create a custom profile field (text field), where people can say if they have been invited by other community member, and write the nickname.

Later, as admin, you can check if that other member exists, and, if so, you accept the new member and let him/her start posting on your forum (playing with permissions)

Is this not what you are suggesting:   https://help.forumotion.com/t146889-suggestion-for-a-full-invitation-system#1010808 ?