This hasn't been my experience. I can get an AC unit and gas furnace installed for something like 7k. But a heat pump unit that heats and cools is something like 13k installed. I obviously went with the former.
I priced out a heat pump versus just an AC and gas furnace in my area, and it was a similar spread a few years ago for me in Texas.
And yet what I see time and again is that initial purchase price is the only focus for construction costs, even rehab work. Those budgets are a separate bucket from utility bills so lower priced projects get approval even if utility bills will be higher.
Anyone doing the math today on a similar purchase should really check out how much they can get from the government: https://www.rewiringamerica.org/app/ira-calculator
Couple "fun" things about a heat pump; You will smell burning the first time the electric strips are used every year moving into the heating season, it's just dust burning off. Once you start moving into colder weather you might notice the system changing into defrost mode so it can melt ice off of the outside coils. There's a variety of sounds that accompany this and you might even see a bunch of steam coming off of the outside unit. It isn't smoke and it's normal. Every year there's a flood of 911 calls about this.