sudo -u <my_username> ./start.sh

That works. For whatever reason. Game screen appears and stays, game seems to work.

(I found that googling around re. Unity/Linux segfaults ... in the (lLinux distribution) Arch wiki.)

One difference I can think of:
If you use the -[root] or -u <user>, it evaluates /etc/profile before starting the game.

Maybe the process does not run under your shell id otherwise? (weird)

That would be basically
source /etc/profile; ./start.sh
Which does not work. (neither does embedding "source /etc/profile" into start.sh - not that I thought it would, but it does not hurt to try)

By the way, another Workaround for People with OpenGL-Problem (especially on Intel-Cards, which doesn't support OpenGL 2.0):

LIBGL_ALWAYS_SOFTWARE=1 ./start.sh

This will start the Game in Software-Rendering-Mode...