The whole idea of these systems to pay for themselves is kind of bullshit. Sure it can reduce on your energy bill, but it will still cost money. Depending on the type of system and the cost to install it can take a long time before any savings start to come.
Remember that expensive systems such as quoted for you are a kind of specialty of wealthy societies and wealthy households. Energy solutions are being sold exceeding their true value, because there is market for that and people still buy. My air-to-water heat pump installation was less than 4000EUR all parts included, although I did the install work myself but it would have been less than 1000EUR for work if I just paid for hourly rates for electrician and plumber; and the end result is way better than how a typical 15000EUR complete one size fits all solution would have been.
Typical cost for air-to-water retrofit was around 8000-9000 EUR here but nearly doubled to 14-15000 EUR almost overnight after a 4000EUR subsidy come into place. The market does not reflect actual costs, and any subsidies further twist the market. Enough people are willing to pay outrageous prices when they feel good about it.
In Japan air-to-air heatpump costs something like 500EUR installed so one can easily afford one per each room. They do pay back for themselves, that's literally why they were developed in 1980's in the first place, to save cost of fossil fuels.
Even when I would get a cheap monoblock heat pump and do the work myself and spend say the 4000 euro you mentioned yours costed, it would still take a very long time to see a so called "return on investment" or reach a break even point.
What I pay now for my heating is 24m3 of fire wood at 55 euro the m3 is 1320 euro.
On average loss of interest on 4000 euro in a savings account lets say 80 euro. (At the moment we get more then 2% here in France, but it will vary)
Needed heat energy 16500KWh at SCOP 3.5 is ~4714KWh at an average electricity price of 20 euro cents comes to 942,8 euro.
So saving comes down to 1320 - 942,8 - 80 = 297,2 euro per year so comes down to about 13,5 years. At that time it might be needed to buy and install a new heat pump. If not you will have to save up the money you save and it will take another 10 years, with interest on interest, or so to get the initial 4000 euro back in the bank.
This is not taking into account the cost of owner ship like maintenance or replacing defective parts if needed, nor is it taking into account the rise in electricity price that comes round once in a while. The latter would also effect the price of fire wood, so not really needed to adjust for.
And that is why I said that living costs money.
It might be different for others who have much higher heating costs with their current system, but still you will never really get your investment back.
To me there is a distinction between saving money and making money. With PV solar panels it might be a different story when you harvest way more energy than you consume and can actually sell the remainder to the energy company. But then still you have to take cost of owner ship into account.