Staff cuts? Yes, plenty - but at least here in western NY, weather staffs are the last thing to get cut. While the news reporters in the field churn every two years when contracts are up (and are lucky if they're making $22k along the way), the three chief meteorologists here in Rochester are all among the longest-tenured air talent at each shop, and I would bet they're paid better than most of the anchors.
I work holidays and sometimes weekends, and have literally never seen an example here in Rochester (market 80-ish) where any of the three stations hasn't had a live meteorologist during a show. There's always backup for vacation or illness. The Nexstar station, WROC, sometimes brings someone in from another market when staffing is short, but there's not a scenario where there's "nothing to fill the weather hole."
For what remains of the aging TV news audience here, weather is still incredibly important in these markets. It's been less than a year since the last mammoth storm that brought Buffalo to a halt. My guy at WKBW slept at the station during the worst of it, when it wasn't even possible to get to a nearby hotel on foot. If a WKBW (or WIVB or WGRZ) deliberately ran days-old forecasts the way you're suggesting, viewers in Buffalo absolutely would be voting with their remotes and going elsewhere.