I think it would be very difficult for anyone to force Kazakhstan regime to increase gas prices by 3x considering they are their own producer

I didn't say that anyone was trying, but ultimately even as a producer if your export market suddenly increases pricing by 3x, the local producers have very little incentive to distribute locally rather than exporting, which will knock domestic pricing too. And the heads of the fuel companies in Kazakhstan are not the people protesting on the streets and bringing down the government.

I do not see how Russia benefits from the fall of Kazakhstan regime.

They don't, it's an unexpected side effect of putting pressure on Eastern Europe and the Ukraine. I imagine Putin expected Kazakhstan to be able to manage this internally better than they have, and are now very concerned. It's something of a nightmare scenario for them.