When I go to the Skins tab in the launcher, I get a message saying "WebGL support needed" (screenshot below). It used to work in a specific Launcher version, but I can't remember what version. However, when I check dmesg, I see the following message:
minecraft-launc: segfault at 7fbb62233c15 ip 00007fbb62233c15 sp 00007fbb5cf916e0 error 14 in libgpg-error.so.0.27.0[7fbb6228d000+4000]
Which makes me think it's not directly related to WebGL.