Brave browser won't open on Linux Mint 19 Mate

Have you tried turning hardware acceleration off? That seems to fix a related issue in other OSs. It looks like this switch should do it from the command line:
brave-browser --disable-accelerated-2d-canvas | Disable gpu-accelerated 2d canvas.
Brave opened and ran and didn’t appear to spew an error.

Here are some related switches that might be worth a try:
–disable-accelerated-jpeg-decoding | Disable partially decoding jpeg images using the GPU. At least YUV decoding will be accelerated when not using this flag. Has no effect unless GPU rasterization is enabled.
–disable-accelerated-mjpeg-decode | Disable hardware acceleration of mjpeg decode for captured frame, where available.
–disable-accelerated-video-decode | Disables hardware acceleration of video decode, where available.

I couldn’t reproduce your problem in Linux Mint 19 (tara) Cinnamon (yes, not Mate, I know).
It’d been a while since I used my Linux virtual machine, so I got to test Brave 0.60.47 in Linux Mint 19 (tara) Cinnamon 3.8.9, kernel 4.15.0.45-generic (#48-Ubuntu), gcc 7.3.0, Xorg 1.19.6, pixman 0.34.0. That worked fine, but I’m using Cinnamon. Even so I updated Mint to kernel 4.15.0-51-generic (among other things) since it needed doing and tried again. Brave 0.61.51 Chromium 73.0.3683.75 still worked in the updated Mint w/Cinnamon. After updating Brave’s signature, got it upgraded to 0.65.118 Chromium 75.0.3770.80 and it seems to be running fine.

Maybe it’s a Mate thing? Admittedly, I’m using a virtual machine in VirtualBox, but it’s now the same kernel and same distro except I have Cinnamon instead of Mate.