How to install NVidia drivers under Linux Mint so that Steam and Proton will work?

Question

I'm trying to make Steam and Proton work for hours with no success. I'm using Linux Mint 19.1 Cinnamon. Whatever way I used to install the drivers, either it did not install or the following situation occurred:

When I start Steam seemingly nothing happens. But actually it launches, and the process keeps hanging. When I start Steam from terminal, it gives me the following:

Running Steam on linuxmint 19.1 64-bit
STEAM_RUNTIME is enabled automatically
Pins up-to-date!
Installing breakpad exception handler for appid(steam)/version(1550534751)
libGL error: No matching fbConfigs or visuals found
libGL error: failed to load driver: swrast

And then the process just hangs there. I can get some additional infos with export LIBGL_DEBUG=verbose, that gives me these additional lines:

libGL: screen 0 does not appear to be DRI3 capable
libGL: screen 0 does not appear to be DRI2 capable
libGL: OpenDriver: trying /usr/lib/i386-linux- gnu/dri/tls/swrast_dri.so
libGL: OpenDriver: trying /usr/lib/i386-linux-gnu/dri/swrast_dri.so
libGL: Can't open configuration file /home/adam/.drirc: No such file or directory.
libGL: Can't open configuration file /home/adam/.drirc: No such file or directory.

/tls/swrast_dri.so indeed does not exists, but /swrast_dri.so exists. I don't think this is really important, because it probably just indicates, that Steam does not see real drivers, so it tries to fallback to software rendering.

I've tried to install different versions, like 340, 349, 418 and probably more, all gave me the same results. I've tried different sources, like ubuntu-drivers, xorg-edgers, graphics-drivers and more. I've tried the official NVidia driver file (that ends with .run), it crashed my X server, I needed to remove it via recovery mode.

I've tried every solution I could find on the internet, no success. I can give you some additional infos that may be useful:

sudo ldconfig -p | grep -i gl.so gives me these:

libwayland-egl.so.1 (libc6,x86-64) => /usr/lib/x86_64-linux-gnu/libwayland-egl.so.1
libcogl.so.20 (libc6,x86-64) => /usr/lib/x86_64-linux-gnu/libcogl.so.20
libGL.so.1 (libc6,x86-64) => /usr/lib/x86_64-linux-gnu/libGL.so.1
libGL.so.1 (libc6) => /usr/lib/i386-linux-gnu/libGL.so.1
libEGL.so.1 (libc6,x86-64) => /usr/lib/x86_64-linux-gnu/libEGL.so.1

glxgears runs fine, no errors. /usr/lib/x86_64-linux-gnu$ ls | grep *libGL* gives me this:

Binary file libGLdispatch.so.0.0.0 matches
Binary file libGLESv2.so.2 matches
Binary file libGLESv2.so.2.0.0 matches
Binary file libGLESv2.so.2.distrib matches
Binary file libGL.so.1 matches
Binary file libGL.so.1.0.0 matches
Binary file libGL.so.1.distrib matches
Binary file libGLX.so.0 matches
Binary file libGLX.so.0.0.0 matches

I'm thinking it's somehow about symbolic links. There's probably a wrong symlink somewhere. Please help me, I'm out of ideas.