I understand why we might board first and business class ahead of everyone else, but is there a specific reason why we don't board passengers back to front? It seems like the better way to go.
As the comments say, the short answer is 'we' don't generally. Airlines use many different boarding systems and boarding back-to-front is generally most popular.
Mythbusters took a look at this and you'll see that there's a problem. Boarding methods that are actually faster make customers less satisfied. So an airline may choose to go front-to-back because it makes customers happier, even if it takes longer.
There are some advantages on the front-to-back boarding are that it gives the back of the plane cabin crew more time to do whatever prep they're doing before passengers start getting there. Perhaps they're still taking the food delivery in the back door? Alternatively you have a crowd of passengers from the front moving down the plane, so cabin crew can move backwards with the crowd assisting. With back-to-front cabin crew are separated at either end by the crowd of people and it's more difficult to assist. Ok, that's just a guess.
Some airlines aim for very short turnarounds and so quick boarding tends to be essential, so they'll spend time and money on research for it. However, for other airlines, it may not matter, so they're free to let people board however they want knowing that they'll probably be waiting for the luggage to be loaded (or whatever) after everyone has sat down.
I do wonder if one particular boarding method is better than another if you had to suddenly evacuate mid-boarding. Might depend how many accessible doors the plane has.
It also all depends on how big the plane is, how much hand luggage people are carrying, how full the plane is, etc, etc.
Basically there are a lot of different ways to do it, all tend to have pros-and-cons but only some airlines really worry about it all that much. For what it's worth I can't ever recall doing front to back boarding, but have seen a bunch of different methods on different carriers.