Taking SpaceX into the realm of publicly traded companies is most definitely a very bad idea, especially with Mars set firmly in Elon Musk's sights.
Publicly traded companies are run by board members who try to pad their wallets as much as possible, and shareholders scream for returns and dividends. Staying private means you can invest all your profits back into R&D, which in a business built around spaceflight is an EXTREMELY important thing. The more money you pump back to shareholders is less money spent developing the next engine or flight control system, or whatever.
And if SpaceX want's to be the first to set boots on Mars (which at this point in time, SpaceX has my bet that they WILL be the first), then once again, being private means you can build the thing, paid for by what isn't being used for R&D or what might be handed over to shareholders.
My hat is off to Elon for building the company to where it is, but it needs to stay the way it is or risk becoming a target for a hostile takeover and being absorbed by one of the existing overweight overly bureaucratic space industry incumbents.
*Flame: Nice looking engine test there!