Congress needs to pass legislation to change COLA. Neither the president nor Congress can ‘approve’ increases; not even gov shutdowns effect SS payments. Talk of SS cuts have been described as the ‘3rd rail’ of American politics for their effect on the electoral prospects of those who threaten such.
On the other hand I wonder if the SS Administration stated that 'if the government shuts down checks will not go out' would really stay the hand of the GOP. The checks go out weekly so half of the recipients would not receive checks in a 2 week shutdown; everyone in a 4 week shutdown. Dems would be able to run on that forever.
Allegedly, seniors on SS will be getting a real cost of living raise this coming January, that is if dems keep the WH and whatever.
Are COLAs automatic?
Yes. Before 1972, Congress needed to approve increases, which it did from time to time. Since then, COLAs have been automatic, though there are occasional years when no increases are granted, reflecting little or no inflation during those periods.