[28370:0203/134512.756947:FATAL:setuid_sandbox_host.cc(157)] The SUID sandbox helper binary was found, but is not configured correctly. Rather than run without sandboxing I'm aborting now. You need to make sure that /usr/share/skypeforlinux/chrome-sandbox is owned by root and has mode 4755.
So I checked that it was owned by root, but the permissions were not correct so I used chmod to change to 4755, and now the program runs. I'm not sure what is the difference between /usr/bin/skypeforlinux and /usr/share/skypeforlinux/skypeforlinux, e.g., the latter gave an error message while the former failed silently. But now that the program is working, they both open the skype GUI. Strange.