Also, at my landing is the actual support rail (not sure of technical term), which is embedded in the actual step...
Is it possible you could take a few photos to show us what you're describing. I'm trying to piece it together but not sure what you mean by support rail. Stringer?
To be honest, I don't think using strip flooring for stair treads is such a good idea, as it will raise the height of each step 3/4". This means your starting step will be 3/4" taller than the rest, and the top step will be 3/4" shorter. Also, now matter how you do it (new treads or on top of extisting) you will have to remove the ballusters, as cutting around them would surely be a painstaking, precission operation. If the newel post is set in the starting tread, you would probably have to remove that as well, which in turn means removeing the hand rail attached to it.
Basically, you're opening a can of worms that could get real complicated real fast. If you're still into doing it, I and the others here will gladly walk you through the process as you go along. But a few photos would go a long way in helping us see what we're up against.