Hi Terry, just followed your steps in PSP8.00 on XP Home SP3 and got the checkerboard pattern with no problems.
In Irfanview double clicking that image and it shows as black rather than green which is how I believe Irfanview shows transparency in a png file (Irfanview 4.30).
Re "getting message layers will be merged" is your layers palette showing more than one layer?
Edit: Used Single color transparency and Alpha Channel transparency with no difference. Always get the green changing to transparent.
Here are my settings.
1. the file I started with - 1 layer
2. Optimiser settings
3. The saved image reloaded.
Edit 2:
Just repeated process in PSP8, X2, X3, X4, and X5 all with the same result. No merging layers warning when saving (as there is only one layer) , reloaded image s all showed transparency as in 3 on diagram.
Systems available Win7, Win 8.1,Win 10 Version 1607 Build 14393.2007 & version 20H2 Build 19042.867