I went through this decision myself a couple of years ago when my old system needed replacing, and what surprised me most was how much the specific setup matters. The headline numbers you see - heat pumps costing $8k-12k versus furnaces at $4k-6k - are real, but they don't tell the whole story. If your house already has ductwork from the old furnace, installation is cheaper. If you need new ducts run throughout the place, you're looking at a much bigger bill. Also, a lot of installers will quote you poorly the first time around, so getting three solid estimates matters.
Here's what I think people underestimate: the backup heating question is real, but it's also not as dire as it sounds. In genuinely cold climates, modern heat pumps still pull heat from the air down to about -13°F or lower, so they work fine through most winters. Where you do run into trouble is if you live somewhere that regularly hits those deep freezes - then the system switches to backup electric resistance heating, which can spike your electric bill for a week or two. That's the gotcha nobody mentions clearly enough. It's not a dealbreaker, but you need to know it's coming.
The payback timeline really does hinge on your local gas and electricity rates more than anything else. If you're somewhere gas is still relatively cheap, you're looking at 10-12 years for the math to work out. If electricity is cheap or gas prices are high where you are, it could be 6-8 years. Either way, it's a long-term play, not a quick financial win. But since a furnace replacement is something you have to do anyway, you're not really comparing "heat pump" to "nothing" - you're comparing it to whatever gas system would've cost you. That framing makes the decision easier.