Just a quick note from a builder, who lives in an area with a 42" frost depth requirement. In most climates with frost, you do not need to provide frost footings for the bottom of a set of steps. Mine are attached to a wide "foot plate", which is a treated 2x10 laying flat. this is laying on a bed of gravel that drains well. This is probably the most common technique in my region, with the balance being stringers that land on a small patch of concrete, like a bit of sidewalk, which is also placed on a bed of free draining gravel. Frost needs three things to do damage. Moisture, low temps, and a media that can store the moisture, like soil. If you eliminate one of the three, no damage.
Doubt me? Every highway in frost country does just fine without frost footers, as long as they are properly constructed on a bed of well drained gravel. A lot of well meaning installers totally screw up frost protected post installations since they don't understand frost lenses, or frost jacking. they dig a tapered hole, with a post hole digger, then pour a cone shaped, jagged concrete pad for the post to sit on. The cone shape and rough edges create places for the frost to grab and jack. Every year the frost will lift the concrete a bit, until the post is pushed up. Once the ground thaws the post fails to settle back down, and the post is a few inches higher. I have seen this process actually push a post out of the ground, over decades of cycling.