Why wouldn’t you just print larger denominations

Because that would be admitting there is a problem. Do you want to be the politician that authorizes change in denomination while simultaneously explaining why you haven't addressed why that is required. I'm guessing the politicians don't want to touch inflation with a 10 foot pole, because solving inflation is painful in the beginning so it tends to be very bad for anyone elected on any timespan less than a decade.

Milton Friedman called inflation like alcoholism, great at the start but quitting is long and painful and most of the rewards of stopping are delayed with only pain up front.

I would add to your excellent answer an anecdote about the tricks of a previous government (with mostly the same people as the current one) to hide inflation: the Secretary of Interior Commerce (a complete thug) forced McDonalds to freeze the price of the BigMac, so that Argentina would look good on the BigMac Index. The prices of all other burgers were more or less free to increase.

So can you actually go in and buy the BigMac? Wouldn't price controls cause inventory of BigMac to disappear? Or does McDonalds happily sell them at a loss?

This was more than 10 years ago. If I recall correctly, they sold the BigMac but did not advertise it (not sure if they even displayed it on the menu).