As to why Lee never competed, only one person can truly answer that question.
Well, more than one actually. I learned more or less the same stuff Bruce did, from the same 'family'.
Bruce was trained to end a fight in one second in a deadly manner, the origins of his art comes from a group of Chinese assassins. It is extremely effective.
There are several reasons that adepts in the deadliest of arts do not compete. As follows:
In competition the players are protected by rules, if most of the effective stuff you know is against the rules then you can't play your game, you have to play the Lumbering Lumox game, where you just stand there and trade punches.
In MMA the players are required to face each other with aggression. In the assassin's arts you either really try to get away, or you pretend to try to get away, this is an extremely important point. There are many tricks which can not be used in the ring.
By break rules I mean things like blinding, + elbow, neck, and back breaking. The rules favor big giant Lumoxes with so many muscles you can't even break their neck, the only chance you would have is to use eye jabs, against the rules.
So, in a 'real' serious art you pretend that you want to get away, act cowardly and groveling, put your hands together prayer fashion (boxer's stance) and beg them to not hit you. If they attack you anyway, in spite of all that, they are telling you they want to die, and in fact it is perfectly legal to kill someone in that situation in the US, as long as you have spectators. There are also certain tricks to use on the audience to make them think that someone attacked you while you were trying to back away, then they slipped and fell down dead. Helps if it goes to court, plus when you blind someone they can't identify you in a police lineup.
Bruce Lee could do all that, absolutely no question, that was actually the foundation of his art. But ending a fight in one second while barely moving does not play well in the movies.