Interesting column. I was told as a kid that it was a sign of prosperity, i.e. “I’m so wealthy and well-fed I can’t even button my vest all the way!”
Not really.
The point is very simple and practical. Vests do reach down to cover trouser buttons/belts, at least they should if tailored correctly. However, they are tailored correctly for people standing, not sitting. Vests should fit rather tight (and in most cases they do, if sometimes from the inside and not by tailoring) because otherwise the suit jackets worn over them would not. If a vest-bearer with the lowest button closed sits down, the entire vest (plus the tie) are pushed upward and either starts bulging at the seams or gaping at the neck.
If, however, the lowest button is open, the vest still covers the belt but when its bearer sits down, it retains its form and position.
What you call a vest we call a waistcoat-a vest is something you wear under your shirt when it’s a bit chilly.
King Edward V11 started the fashion for leaving the bottom button undone.Doubtless it was done for comfort as he was slighly on the stocky side-his nickname was ‘tum-tum’. We ludicrously class-conscious Brits followed the fashion.
He was so fat that on his visits to Paris the brothel he frequented made a specially designed chair for has assignations-it was featured on a BBC tv programme about the brothels of Paris
IIRC, as a kid I was told it was a thing you do inside, like taking off your hat, and that you re-button it when you leave a building.
Some other adults in the conversation said it was more akin to a relaxed posture, like loosening your tie after work - you unbutton the bottom button because it’s more comfortable.
Along the “nuts and bolts” reason, I also recall that if your vest is snug (but not “tight” as in you can’t sit down) before a meal, it will be very tight after a meal. So, sometime during the meal, you unbutton the bottom button to make some room.
Finally, I think it might be something you’d do if you’d put on a few pounds since the last time you wore the vest, and you can’t button the lowest button 
This legend is acknowledged in the original column. But do you have any authority for it? (“My mother told me so” is not evidence.)
I have just been looking at a website called ‘the fedora lounge’. There’s a picture of Edward VII in a waistcoat-with the bottom button done up!!