Conflicting contractor opinions on rebuilding steps using original construction method
Some thoughts:


Single stones that size will be ridiculously heavy.  Consider using three stones per step, similar size stones for the landing.
You don't need solid brick under the stones, but you need something.  You could compact fill dirt for bulk filling, then a layer of paver base, and top with a layer of sand under the stones at each level.  If you really wanted to get fancy, you could run a couple of rows of brick supports lengthwise, so they would support the seams if you use three stones per step (and fill as described in the cavities).
In terms of dimensions, the steps should follow a ratio of 2 x rise + tread = 26 to 27 inches.  If those are standard bricks, your dimensions will not feel natural, and people will have trouble walking them with a normal gait.  It would be closer if you rotate one brick on each side of the lower steps so that the tread is 12" instead of 16", then add a brick to the landing to maintain the total length if needed.  But it's only three steps, so it may not be a big deal.