My guesses: he chose to donate to FLI because their star-studded advisory board makes them a good public face of the AI safety movement. Yes, they are a relatively young organization, but it looks like they did a good job putting the research priorities letter together (I'm counting 3685 signatures, which is quite impressive... does anyone know how they promoted it?) Also, since they will only be distributing grants, not spending the money themselves, organizational track record is a bit less important. (And they may rely heavily on folks from MIRI/FHI/etc. to figure out how to award the money anyway.) The money will be distributed as grants because grant money is the main thing that motivates researchers, and Musk wants to change the priorities of the AI research community in general, not just add a few new AI safety researchers on the margin. And holding a competition for grants means you can gather more proposals from a wider variety of people. (In particular, people who currently hold prestigious academic jobs and don't want to leave them for a fledgling new institute.)