You don't have the C wire brought out to the thermostat.
See that COND UNIT CONTACTOR On the right is the Y wire going to the stat
On the left is the C wire going to the transformer common. It is not labeled C but nevertheless it IS the C wire.
If you have more than 4 individual wires going to the stat then you would have an unused wire, you could connect that to the left terminal of the COND UNIT CONTACTOR and get your C wire brought out to the stat
Or you could run new stat wiring.
Or you could wire in the C wire adapter. Usually how those work is they take some of the signaling wires like G, W or Y wires, convert them into a multiplexed set on a single wire, then the thermostat breaks out the multiplexing internally. Then you repurpose the now-unused G W or Y wire to a C wire.
By the way if you live in a temperate climate then the heat pump isn't worse than a gas furnace. I'm like you, I am old-school with a central gas furnace and a central air conditioner. But I'm in a temperate climate and a lot of people around here run heat pumps and their costs are about equivalent to my combined electrical/gas.
Where heat pumps really suck (in my opinion) is cooler climates where you get a lot of snow and such in the winter and it's not that hot in the summer. Such as like Montana 50 years ago before Global Warming scotched everything.