I made sure the store page had the "controller support" tag.
The controller works perfectly with every other game including older games and indie games that have some compatibility problems (like not showing the gamepad button prompts and showing the keyboard buttons instead).
Tried closing Steam and restarting (I don't have it on "launch on Windows start").
Only other solution I found was to install third party software which I'm not willing to do.

Game looks good so, if I have to, I'll resign myself to play with KB/M but if someone knows a reasonable solution that doesn't imply installing other software I'd appreciate it.

Thanks in advance.

This game came out around 5 years before the DualSense, so your request is absurd. However, this is the PC and there is a way to play with it and even with the PlayStation icons since the game has compatibility with the DualShock 4 natively.

You must install DS4Windows, make a Profile and there in the subtab Other > Virtual Controller Settings, select DualShock 4, Save it, then in the main tab Auto Profiles Add the Mad Max game and select the created profile in Controller 1. Now you just have to start DS4Windows and turn on the DS before the game starts.