Based on my many years of competitive play, I echo the response that an exclusively random strategy is not only unworkable but illogical.
In the first place, you are correct that in theory, the strategy will win 50% of the time and lose 50% of the time, excluding ties. While it is true that a batter hitting .500 will lead the league, RPS is quite a different sport. No RPS player would be very happy winning half of his or her matches. The path to winning a major tournament is narrow and steep. The winner of, say, a 256 player tournament will have won eight consecutive matches. Clearly, a strategy that only yields a win every other match will be useless in getting one past the second round.
Consider also the gold standard of competitive RPS: the best two of three. Again not counting ties, if a match goes one and one before a resolving throw, the final score will be 2-1. In percentage terms, this is roughly 66% to 33%. How does a 50% win rate fit in here? If one is tied on a match, one wants to win the next resolving throw, not one of the next two.
Perhaps the clearest refutation of the exclusive random strategy is to point out the weakest link in any system: the human element. Let us assume that one takes your suggstion, gets a computer to generate a random string, and memorizes the results. It is conceivable that an ill-intentioned associate could take it upon himself to steal the list and sell it to one's opposition. Even were one to destroy the hard copy, it is possible to have one's computer pilfered in a search for such information. (Like most professional RPS players, I have had my training logs stolen as a matter of course. Like most other players, I always leave a few "decoy" folders out with counterinformation.)
This is not to say that random (or pseudorandom) strategies do not have their place. All players should be able to switch into random mode when the situation calls for it. Some players are more succeptable to random play than others (for instance, many of the great reactive players get stonewalled when faced with a non-reactive strategy.) I also find that a pseudorandom strategy is a great transitional strategy, seamlessly bonding together, say, an Avalanche and an alternating paper-exclusive stance. During his competitive years, Wojek Smallsoa was famous for this, his daring attacks coming as a thunderbolt from a clear sky.
I have played against my share of computer programs in my time, and find all of them superior to a random throw generator (RTG.) In exhibition matches against RTGs, I find that I can more clearly focus my intuitive and precognitive abilities, without having to worry about psyching out a human opponent, or defend against being psyched out by them. There is no need to cloak one's throws or pay attention to an opponents perhaps deceptive physical tells. In all important ways, I find exclusive random play to be vastly inferior to the complex adaptive systems that were my signature style during my competitive years.