Chris, you really wanna remove all that half-inch subflooring. Especially since even the good parts are likely not in real good condition.
The very best you can call a half-inch first layer over joists is a spacer.
And you don't sound like you need a spacer.
Cut it all out and start at the joists with a minimum of 3/4" T&G subflooring or square-edge 3/4" with blocking at the seams.
Then, as Davy suggested above, it would be best to add another layer before you tiling substrate and tile.
But you can, technically, install your substrate directly over the single layer of 3/4" subflooring if you want.
My opinion; worth price charged.