Xbox360 managed both. You either plugged in the pack with two AAA’s or you could use the one with a rechargeable battery that fit the exact same slot.
Seems a good compromise to me.
Meanwhile, Sony went with an internal battery but had a standard USB connector.
Nowadays, you can get an Xbone controller and still choose a replaceable battery pack or a rechargeable one, and it has a USB-C connector. Standard connector, choose of battery feels like a good way to go for me.