I hope you are all okay, thats sounds like a pretty brutal accident. I've had stairs let go under me, thank god I didn't have a fridge follow me down.
As you found out, stairs are usually well used and can face significant loads. Carrying a fridge down or a 13 year old jumping down three steps at a time puts some heavy loads on a step. If you're confident with your carpentry skills and are familiar with (or can learn) how they are built properly, go for it.
However, it not a job to take lightly, if you have any concerns or there are liabilty issues (fridge on second floor sounds like rental property?) it would be money well spent to hire a pro.