Let's approach the answer to your question from a pure trading and risk management perspective because looking at it from a mathematical standpoint nor quant standpoint does not yield you much here:
1) Bollinger bands are nothing else than standard deviation envelopes around the mean of past prices of the underlying. So, as far as simple probabilities go, the bollinger bands are not supposed to be broken more often on average than the volatility of past price moves indicates. Therefore, nobody should be surprised to see assets to trade most of the time in between their 2 or 3 standard deviation wide bands (only difference here is that the bands are around the mean and not the actual current price).
2) "Research points to price changes in stocks as a random walk. Also, most people believe that past news, and other factors have little if any impact on future price movements.":
-> First of all stocks nor ANY other asset follows a random walk. A random walk is as close as the lazy academician gets to modeling stock prices. It is far from being accurate. Its the same as saying interest rates are constant and volatility has non random components when valuing option prices with Black Scholes. A random walk is an incredibly weak tool in my opinion to model stock prices. Actually a very often asked question to aspiring junior traders is whether they believe in efficient markets. Believe it or not but a huge number of highly accomplished quants who want to move on to trading desks do not make it because they answered yes in one way or the other to above question.
-> Past performance, price movements, volatility behavior is one of the most often used and most important component in ANY pricing, modeling, trade evaluation and risk management approach. Care to forecast volatility without past datapoints? Care to make an educated guess about buying or selling stocks only by knowing its fundamentals? Care to make markets in options without knowledge of previously traded levels (you will lose your job faster than you can blink with your eyes)? My point is that its one thing to comfortably sit in your quant seat and model time series with KDB and price up your exotics using Monte Carlo or other non-closed form techniques and hand it over to the trader. Its an entirely different thing to be in the hot seat day in day out and show prices worth sometimes hundreds of millions, knowing the market is sometimes HIGHLY INEFFICIENT which was sadly an often forgotten but very important risk aspect on quant desks. I am not trying to rant against quants but I simply hugely disagree with many quants that past prices are close to being irrelevant, such belief would get you instantaneously fired from any trading desk.
3) My short answer to your last paragraph is, it depends: The only thing I agree with one quant who also submitted an answer here is that you need to rigorously test your ideas especially if they can be tested. I am willing to bet there are people who make a killing trading one or the other variation of bollinger bands, and there are on the other hand 1000s of others who lose money day in day out with such approach. Why? Because some are willing to put in the work, test, refine, test, start over, test, verify, trade small, build, re-test, optimize, add size,.... Most are not willing to do that. That is the simple but my honest answer. Those who spend the effort to test and verify ideas and are willing to risk money on promising strategies will be rewarded, others not.