That's true for gas fired plants so I expect it would be the case for light liquid fuelled ones. You wouldn't burn petrol in them, because that fuel is very specific to piston engines (turbines don't need toxic anti-knock additives, for example).
As far as grid stability though, in more advanced countries it's relatively easy to build out more grid capacity, just expensive. Only in politically troubled countries like Australia and the USA does the "but how do we create an effective market for building the grid" dominate that discussion, and result in a poorly functioning grid but also (obviously) a badly broken market in providing a grid. Back when I was exposed to economics electricity grids were a cliche example of things where markets are inappropriate and inherently unworkable - they're a natural monopoly. Even the most die-hard capitalist isn't going to suggest two or three independent grids supplying every building... well, maybe in Texas :)
I'm in Australia, and the big advantage we have over much of the USA is being closer to the equator... only about half our current houses have solar PV on them, and less than a third of commercial buildings. We could get 20% more electricity just by fixing that, and it would be conveniently close to demand as well.
In that case, you could just skip the grid and put the turbine in the car.[1]
Yes, the vehicle is heavier (=less efficient) when you carry the range extender around with you, but that's compensated for by not needing as heavy a battery. Eliminates dependence on the grid; replaces it with dependence on the current refuelling infrastructure. Which latter, at the moment and in most places, certainly is better than the currently available EV charging infrastructure. Crossover point where pure EVs become more efficient TBD by how fast fuel infra is reduced and how fast charging infra is improved... With an additional confounding factor introduced by GP's hypothesized grid instability.
