So, two questions. Would you use good old yellow glue or would something
else be better suited? Screws or no screws? Thanks in advance for the
wisdom.

Thanks for the advice. Confirmed my thinking as to the screws. Like the
idea of epoxy, but since this is the second rung I've had to replace, I had
better not go that route. Looked at this stools sister and it was also
assembled with screws - and guessing they did serve the purpose of clamps.
I'll ask the Amish family next time I go and get wood and see what they
say.