Saying Before Christ confers on him a title that non-Christians would not feel he deserves (calling him Christ recognizes the claim that he is the Messiah). Similarly, saying "In the Year of Our Lord" is problematic if you don't consider him your lord. There's a real difference between using Jesus' birth as the break in the calendar and having the terms you use for the calendar recognizing a status for him that one may not be willing to grant him.

I'm an Atheist but use the BC/AD abbreviations, merely out of habit and personally never felt the discussion of using BCE/CE or BC/AD was really worth having.

So what I'm here to actually say is simply that the year 0 doesn't exist in either system. The so called "year of the lord" would be 1 AD, though you are correct that his likely birth year was a couple of years (4 to 7 if memory serves me well) earlier. In other words our calender goes from 1 BC to 1 AD skipping a 0 AD/BC.

It's one thing for somebody to accept the calendar which enjoys the greatest use worldwide, and which is overwhelmingly dominant in the English speaking world. It's another thing to ask non-Christians to refer to recent dates with the phrase "in the year of our lord", which is what Anno Domini (AD) means. BCE/CE lets non-Christians use the calendar that, conveniently, everybody else uses without making them feel like they are endorsing religious beliefs they don't espouse.