My 2 year old has her own Kindle Fire for kids. And I don't give a shit what anyone thinks about that 
I have an expensive iPad that I don't want breaking it covered in sticky fingerprints. The fire comes in a child friendly case and is replaced by Amazon if broken by a kid so no issues there.
It is fab for long car journeys, entertaining her (on mute) in restaurants if she is restless, and yes she watches the odd cartoon on there. She mainly does puzzles, jigsaws, number/alphabet games and 'paint' games. She knows much of the alphabet and can count to 20 through playing games together on it,
She doesn't mither to use it, to her it is just another toy and doesn't make a fuss over it. She probably spends about a maximum of hour a day on it, spread out in little bursts of ten minutes here, five minutes there etc...
She has tons of books and toys, craft stuff, paints, play doh, dressing up outfits, everything, we go on walks, to playgroups, she isn't deprived of being creative or learning through play.
The world we live in today is full of amazing technology and we should embrace it, not act like it is something to be ashamed of if you let your kid use them!
Each to their own, of course, but I personally don't understand all the 'screen time' fuss. As long as it is handled age appropriately and they there shouldn't be an issue.