Can these grants provide a living wage?

Wages paid by grants in general have to match whatever the institutions 'standard' pay scales are. So it's the institutions deciding pay rates. At least in the U.S..

My wife is currently in the middle of a fight over this. Her salary is paid by a grant from an organization that has a minimum pay scale for postdocs. The university has its own standard pay scale for postdocs, which is about $20k/year lower. So far they've refused to budge. It's not even their money, there's just 4 layers of bureaucracy who have nothing better to do than ~enforce policy~ no matter the cost.

Of course they'd fight your wife. They don't want any history or evidence of potentially paying more under any circumstances.

Perhaps so, if universities cut back on overhead. Much of the grant funding is taken away to pay administrators who don't directly contribute to research. There is a huge amount of waste and no real pressure to control costs.

Depends on where you live: https://csstipendrankings.org/

NSF grants are hard to get, with a small award being for about $600K over 3 years. That translates to about $300K after the university takes its overhead. For me, that would about fund about two PhD students for 3 years for about $31K per year during the 9 month academic year. Many PhD students in computer science do internships during summer where they make a lot larger salaries, or they are funded by me for that period instead.

In my city, Rochester, according to this website, our stipends are the most generous in the USA based on the low cost of living here. https://csstipendrankings.org/

We would all like to pay students more, but either we then have fewer students or need the government to award larger grants. As someone who had a hard time getting into a PhD program due to how competitive it is, I would be opposed to depriving folks of educational opportunities.

Students typically take classes and do little research for their first 1-2 years, but they do receive a stipend during that time period.

Well, if we got rid of the administrators taking half the grants, we could probably almost double the wages. So in many cases, yes.