You will confirm the idea that there is no oily polish, etc on the steps if the sandpaper does not gum up. The alcohol will clean off possible contaminants and remove the fine layer of dust that the vacuum can not. In my opinion, TSP is very soapy and would take volumes of water in order to know it has been completely removed. Keep your cleaning rag folded in squares and flip sides when it starts to get a little dirty. Keep the rag plenty wet. The alcohol will dry immediately so you can varnish right away.

Use 2 inch blue tape on the carpet, lifting with your fingers to tuck the tape under the carpet. Try to go completely under the carpet edge so you can run your poly under the edge just a bit. You may find some polyurethane just for floors at the big box or your local paint store.

When you are done, it will probably look better than it ever has.

Best, CPeace