"No matter what you do, statistically, on average when BRT is used you will check a 0. You have exactly the same chance to have an attack be revealed by BRT than using an attack as that check."
Did you think about this for more than thirty seconds before posting it? If you reveal an attack, there's now one less attack in your deck and there are (probably) more foundations than attacks left in your deck, so if you reveal a 2-4 initially, the odds are much higher that you will check >0.
Run more 2-4 and a handful of 6 checks and BRT will have significantly less impact.