Wanted to use a proper controller with your phone? You've got all the choices with Android 10 now, with the OS supporting every major console controller.
The commit has a key layout, allowing the Android OS to properly recognise inputs from the Pro Controller. It's currently only accessible via Bluetooth, and if you've got Android 10 on your phone you can connect the controller by hitting the pairing button next to the USB-C port and pairing the device through the Android Bluetooth menu. You can even use the controller to unlock your phone once it's paired (by adding it as a trusted device in Android settings).
The addition means Android 10 now supports basically every major first-party console controller, with support having already been added for the DualShock 4 (which lets you use the touchbar as a mouse), the Xbox One S revision of Xbox's stock pad, and the first Xbox Elite controller. There's also a wealth of third-party options, like the capable 8BitDo SN30 Pro+.
Support within games is another matter entirely, although games like Fortnite, Call of Duty Mobile, all the SEGA Forever games, Crossy Road, Evoland 1 and 2, Stardew Valley, Horizon Chase, all support gamepads on mobile. It's worth noting that you can use PS4's Remote Play through Android with a DualShock 4 as well, if you're looking for another gaming option on your phone. (Might be worth considering for Death Stranding, if you want to have something on in the background.)
Is it the best way to play some of those games? Not always, although from an accessibility standpoint the option is nice to have. And Call of Duty is easier with an actual controller as opposed to a virtual one, even though I'd still prefer a mouse and keyboard.