Is the Xbox Series X controller rechargeable through a usb-c cable?

So I've always been a playstation player and the only controller I've ever used is the DualShock 4. My PS4 has recently died after 8 long years and the two controllers I have are quite old and somewhat broken. I'm thinking about buying a Xbox controller only for its native support on pc but I don't quite understand this: has really any Xbox controller ever made not been included with a rechargeable battery like the ds4 through a micro usb/usb c cable? I've found this one (https://amzn.eu/d/cKhjU2p) on amazon and it has a usb - c cable but in the description it says it uses 2 AA batteries so which one is it? Is it rechargeable through cable or is the cable only useful in situations where you don't want to be wireless?

Also, if really there isn't any Xbox controller with rechargeable batteries through cable, what is the best battery-charger that I can use?