Below is linked a custom launcher to bypass the original launcher. The launcher itself requires AVX instruction set support, but the games themselves do not. TLDR: If the launcher is not running it may be due to old CPUs not supporting the AVX instruction set. This leaves the custom launcher largely obsolete, but nonetheless I will leave it up in case anybody still has a use for it. The 7 June update for Mass Effect Legendary Edition removed the dependency on the AVX instruction set in the launcher, so this custom launcher should no longer be needed for people with older CPUs. See the NexusMods page for new version and the list of changes. Some people are still using it to unlock all of the language options that aren't always available in the original BioWare launcher.
EDIT: Since the 7 June Update I have updated the custom launcher to v2.1.