Similar experience with our 2yo & YouTube. It is banned in our house for now, because 2yo doesn't exactly understand moderation, but we are willing to open it back up as they get older and understand a bit more.
Check out the Amazon FreePlay system. I got a "kids edition" kindle for my 2 year old, it came with a year of freeplay and I've kept it going since it's such a good value.
Basically, they get access to all the kids apps, books, and videos in the kindle store. You can limit the age appropriate level.
My biggest complaint is how tedious it is to remove apps, since the kids eventually fill up the kindle.
Good for you. Our 5 and 7 year olds get "banned" from youtube every now and then when we detect the slightest bit of "mania", which they do get.
In Sweden no screen time is recommended until the age of two, and I would easily say that until 3 or 4 years of age, screen time is really not necessary and possible harmful.
My biggest complaint is how tedious it is to remove apps, since the kids eventually fill up the kindle.