Installing d3dcompiler_47 with protontricks and setting the renderer to directx works if:

• You have Mesa 22.0 or higher (Proton Experimental bumped minimum version to Mesa 22.0)
• You are using the newer Intel driver. Mesa 22.0 doesn't work with the classic i915 driver

The problem

Linux Mint 20.3 ships with Mesa 21.0 and it uses the classic i915 driver by default. Mesa 22.0 does not detect the required DRI3 support from i915 since support is deprecated.

Solution

1. Install the Kernel 5.15 since it's required by Mesa 22.0
2. Upgrade to Mint 21 and/or install the latest Mesa drivers using this PPA: ppa:kisak/kisak-mesa
3. Enable the latest Intel driver and DDI3 in the XOrg configuration:

Section "Device"
    Identifier "Intel Graphics"
#    Driver "i915"
    Driver "intel"
    Option "DRI"   "3"
EndSection

Once you restart your system you should see that DRI3 is enabled:

% cat /var/log/Xorg.0.log | grep "direct rendering"
[  1428.211] (II) intel(0): direct rendering: DRI2 DRI3 enabled

Some useful commands

Detect mesa version:

glxinfo | grep "OpenGL version"  
OpenGL version string: 4.6 (Compatibility Profile) Mesa 22.2.0 - kisak-mesa PPA

Detect if hardware acceleration is enabled

glxinfo | grep "direct rendering"                
direct rendering: Yes

Detect if DRI3 is really enabled

% cat /var/log/Xorg.0.log | grep "direct rendering"
[  1428.211] (II) intel(0): direct rendering: DRI2 DRI3 enabled