Is this just a dumb idea (my gut tells me yes) or is it pretty stable?
Stable? Definitely not. (There are random uneven gaps between the bricks, both vertically and horizontally. They will move if there is any stress.) Does it matter? Not if the steps and landing are well made and self-supporting.
Multi-storey buildings usually have stairwells where flights of concrete stairs have no supports except at the landings (top and bottom).
From the picture it appears that you're talking about a considerably shorter span, even including the landing outside the door, so as long as the landing and steps themselves are well made, and they're supported on at least two sides, what's under most of them is irrelevant.
I feel obliged to point out that "well made" would mean properly strengthened with a consistent thickness.
- It should be reinforced (with rebar, mesh, fibre, or some combination; and in an approved pattern)
- Check your building code authority for the required minimum thickness (which may depend on the span and maximum load and other factors) and add a good margin.
- The top surface of the landing should be below bottom of the door by a sufficient margin that it doesn't act as a catchment to drain water into the house. (If necessary remove bricks so that there's enough space to allow this AND the minimum thickness.)
- Putting rebar in amongst the bricks sounds wrong; you want the reinforcing in the concrete itself, not under it.
- The outside edge all around should either
- be formed into a reinforced "beam", supported on "posts" on the corners; or
- go all the way down to the footing (which itself should go into the ground).
Consult with a real builder for the exact requirements.
I suspect a better use of the bricks would be to form the sides, as long as they're the type of brick you can put rebar through to keep them aligned, and partially filled with concrete.
Or of course, just sell them.