I hear this very often, but has this ever been proven?

Yes, multiple times we've seen this play out in real life. The most recent big one would be Japan 1990s. Look up "The Lost Decade". The biggest one in the USA was the Great Depression in the 1930s.

People are not exactly going to stop eating or paying rent which already eats more than half of people’s income.

You stop buying all but the absolute basics (bulk beans and rice and nothing else?), so all the food that isn't a "basic" goes un-bought and the people producing that food/product lose their jobs. Then you stop buying food when you're out of money, which is what can happen in an extreme deflationary environment because you have no income, because you lost your job, because no one buys whatever your work produces.

And if you stop paying rent, you're going to get evicted. If you stop paying your mortgage, the bank will take your home. This played out exactly this way during the Great Depression.

I could see gambling and entertainment become more stale, but I’m not sure how big of a problem this is.

Its much more than just those sectors, but just imagine how many jobs are in the gambling and entertainment business. Now all of THOSE people are out of work competing with you for whatever jobs remain. That, too, happened during the Great Depression.