Quote The times that ball is sided in are against other aggro decks, and against combo decks where you need as quick a clock as possible. Most aggro and combo decks have such tight mana curves that they end the early turns of the game tapped out so one of ball lightnings main drawbacks, it's vulnerability to spot removal almost never comes up.
Also, you can often produce RRR if not by turn 3 then by turn 5 and the 6 points of damage can often act as a finisher or at least put them in burn range.
This view on Ball Lightning is flawed. You're forgetting that Sligh isn't the only deck that runs burn spells - R/G Beats and Zoo run burn (and some Zoo builds run Swords to Plowshares as well). On top of that, you have TnT and Suicide Black running Masticores. Even the Fish archetypes have weapons to take out Ball Lightning - Fire/Ice, Razorfin Hunter, Sea Sprite, Psionic Blast. As far as paying RRR, I'd rather spend it on 3 3-for-1 burn spells instead of a vulnerable one-time use that could easily be taken out.
Ball Lightning isn't good against combo because combo doesn't care how much damage is dealt to it as long as it can go off in enough time. Therefore, spending RRR on 6 damage turn 3 is not good, and even by turn three it may be game over already. Casting a Ball Lightning on turn 5 is not good either because by then you should have been combo'ed out or you should be able to win without the Ball Lightning.
Quote Browbeat puts your opponent in a quandry as either choice they make is very good for you. It's either a 3 mana lava axe or a 3 mana ancestral and usually the next 3 cards contain more than 5 damage (barring drawing multiple land). Against a deck that cannot mana drain it (keeper, Urphid, Mono blue) and cannot race it (academy, WGdragon, Mask) it's golden.
Here's the thing though - You're saying that Browbeat won't be good against quite a few combo and control decks, therefore making it useful against only the aggro decks. Now, if you're the aggro player, you would most likely take the 5 instead of letting the Sligh player draw 3 cards which could potentially be more of a threat, correct? Therefore, this is a 5-for-3 burn spell, which really isn't making the best use of 3 mana, is it? It can't remove opposing creatures, and it most likely won't draw you those three cards that could make things worse for the opponent. Wouldn't there be better choices to put in the sideboard then to deal with aggro decks, like Pyrokinesis?