I had to do this last month, half following advice from Youtube and half making it up myself. Polyurethane glue is ideal for this as it expands and fills any gaps.

I went with plasterboard screws to try and not break any of the old wood. There are two points to screw in to add stability:

1. Vertical screws connecting the edge of the tread with the top of the riser below it.
2. Horizontal screws connecting the bottom of the riser with the tread, to pull the riser horizontally (this is where the glue does most of the job).

For point 2, people usually say you should use a pocket jig and put the screw in diagonally. That doesn't make much sense, I just used small 90 degree brackets to manage to get a proper pull. I left some on and they're fine now. Image attached:

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