The Problem:
Obviously this is a Windows game, I tried different Wine and Proton settings in Heroic, Lutris, and Steam, but couldn't get the mouse to click on the buttons at the main menu. No graphical problems, only that the mouse cursor wouldn't click on the buttons.
I did read somewhere that an additional check was added to see if the game window was clicked or not, and the solution involved some weird hex editing and DLL changes. But I'm not a fan of downloading a random software to modify DLL settings and whotnot.

The Solution (for me at least):
I was able to get the game to realise I was clicking the buttons by:
- Installing the game with Heroic.
- Install ProtonUp-QT and ProtonPlus (thanks to Robo n' Tux Guides on YouTube for that video)
- In ProtonPlus, select to install Wine-Lutris -> lutris-wine-7.2-2 (or I assume the latest one you see)
- I didn't use ProtonUp-QT to make any changes, but too afraid to remove it now :p
- For the configuration, I deleted all prefixes and started from scratch, so the below is only the changes made.
- In Lutris, configure Wizards & Warriors with the following settings:
- Game info -> Runner = Wine
- Game options -> Executable = (...your path/Wizards and Warriors/deep6.exe)
- Game options -> Working directory = (...your path/Wizards and Warriors/)
- Runner options -> Wine version = lutris-7.2-2-x86_64
- System options -> Restore resolution on game exit = true/on
- System options -> Switch resolution to = 1024x768

For the ones that want the tech specs of this fossil I play on:
- Linux Mint 22 Cinnamon
- Intel Core i5-8400 CPU @ 2.80GHz × 6
- 23.4 GiB RAM
- NVIDIA Corporation GM107 [GeForce GTX 750 Ti]
- X11
- (yes I had to kill that nouveou thing and use proprietary drivers, if it's relevant)

Edit: Keyboard input didn't work (I got to the naming the character part) until I alt+tabbed out and back, then it worked (as suggested by yet another random forum post somewhere).