You have good colour vision. We seem to be in the same age group, so it is good to confirm that you can distinguish the hues quite well.
It's much more than the software. It's a combination of the abilities of the hardware, having the hardware that is properly set up and the working environment. It is definitely not just the software.
Tt depends on what is in the image. If you photograph brightly coloured flowers or tropical birds with brilliant colours or shoot night scenes with bright, intense lights, then Adobe RGB will display colours "more accurately" than an sRGB screen. The wider colour spaces handle the brighter and more intense colours. AdobeRGB can handle about 50% more colours than sRGB can. If those colours are in the scene and they are in gamut for the wider gamut hardware, then an AdobeRGB compliant screen will render the colours more accurately.
That being said, those colours have to be in the image for this to be important, so if you are shooting subdued colours, chances are that the entire image might not be as wide as even the sRGB colour space, then there is no advantage to a wider gamut screen in that particular situation. Both an sRGB and AdobeRGB screen will replicate the colours accurately.
People would not spend the extra money on an AdobeRGB compliant screen if it did not make a meaningful difference in how well the colours are reproduced. I use a dual screen setup with my primary screen being AdobeRGB compliant and my secondary screen being an sRGB one. There is a perceivable difference in how the colours look when I have a scene with colours that are OOG for sRGB.