Is purchasing a heat pump really not worth it?

I have always been a supporter of eco-electricity and renewable energy sources, which is why I wish for an air heat pump for our own house. My husband is much more conservative, and we are currently unable to find common ground. Based on the current gas prices, he calculated for me that an air heat pump would only pay off after 15 years. He assumes acquisition costs of around 10,000 euros. But is this rough estimate/calculation really correct? 15 years would also be too long for me..... We have a solid construction with 17 cm Unipor bricks and 14 cm WDVS

If you are interested in an exact price for a heat pump, have one offered by a heating engineer. However, the price alone is not sufficient; the heat pump must be precisely designed for your house.
Regional weather data is also part of the thermal insulation of your house in order to determine a heat pump with sufficient performance.
It would be advisable to seek advice from a professional engineer; although this costs a fee, you would have professionally determined data on heat supply with a heat pump, costs, and amortization.

Your husband's price estimate is correct, however; you have to calculate within this price range.

would have thought that a heat pump only makes sense from a certain size of the house

That is also what it does and therefore consultation with a specialist is necessary. This includes detailed calculations and which heat pump can be used. A cost estimate is often also provided.

I also sought advice and it is all too expensive. It will only pay off after at least 15 years and that is a long time. So if you have to watch the money, a heat pump like that is simply too expensive.

Heat pumps are relatively expensive to purchase, that is true. The other question that arises for me is, with a heat pump I do not need any other fuel. I only need electricity to operate it. This makes me relatively independent with this energy source.