Let's say that you buy an $800,000 place at 3% interest with a 20% down-payment ($640,000 borrowed). Your payments are $2,698/mo. Fast-forward a couple years and you're now at 7.3% and your payments are up to $4,388/mo. That's a 63% increase in your housing budget. That's an extra $20,280/year in housing costs.
Yes, rent can increase crazy amounts which makes budgeting hard as well and we've seen rent go up 20% from pre-pandemic levels in many markets (though that seems to be rapidly falling as the market changes). However, you're not committed to that rent. Yes, it might be bad to downsize to something smaller that you're renting or to a less desirable location, but it's theoretically possible. In this case, you now have to pay 63% more money without a possibility of alleviating that burden.
As rates go up, the amount you can sell a property for likely goes down (since it's now a more expensive property). In the US, at least you can continue living there at your low interest rate and fixed monthly payment. In NZ, your $800,000 place is now worth $700,000 and your payments have gone up 63% and you can't really sell it because you owe more than it's worth and you also can't afford the payments...?
I feel like I might be missing something, but your answer might just be "yea, that's how it is here."