Maybe that's true in the US (I don't know), but it's certainly not true elsewhere in the world; such as Australia for example where most home loans are have a variable rate.
No, never massive swings in the short term (GFC caused a 4% drop (7% to 3%) in 2009 for example). I think that even when the cash rate is moving quickly rates don't change more than a percent or two per year.
Re budgeting, I think the relatively slow rate of change makes it mostly a non-issue though I think people at the upper end of their borrowing power struggle with rate rises.
The ARMs (Adjustable-Rate Mortgages) here in the US can jump pretty severely, mostly because they are only given to folks with bad credit. Folks with bad credit push to buy something out of their budget, get behind, rates spike, then lose it all to the bank, continuing the bad credit cycle.
How often do the rates change? Anything crazy like 2-3% swings? Doesn't that make budgeting tough?