Heat pump tariff with us: 19.91 cents; gas tariff: 5.77 cents. Basic fee comparable in each case. With an annual performance factor of 3.5, the air-to-water heat pump does not save anything compared to gas. For the air-to-water heat pump, also note the inspection requirement according to EU Directive 517/2014, or if the currently desired heat pump does NOT yet fall under this inspection requirement, please assess yourself whether there might be changes in the future. In comparison, the chimney sweep every two years is a joke.
Another interesting topic that Prof. Leukefeld has occasionally mentioned: heat pumps consume electricity especially when it is expensive (many consumers in winter, additional heat pumps will increasingly raise these consumptions). The trend is towards "smart" electricity meters, and anyone who adds 1 and 1 should soon expect that winter electricity will eventually become more expensive than summer electricity.
Then there are the possible hygiene problems (the air-to-water heat pump will hardly be able to temper the hot water tank to 60 degrees), which can in turn be compensated by other costly systems (hygiene hot water tank) and the possible problems with noise and neighbors as well as the aesthetics.
Gas: is used by about 50 percent of all Germans for heating; an affordable gas price is therefore politically highly significant, and the gas consumption you have in a new building can heat 1-2 rooms in other cases. The market price for gas is currently dropping sharply; not even all current price reductions have been passed on yet. Secondly, there are numerous methods to produce gas alternatively, whether so-called wind gas, biogas from plants, or biogas from manure. At the moment, gas is even the only conceivable seasonal storage for the energy transition. Considering the renewal rate in existing buildings and the fact that even then you cannot simply switch to alternative heat generators, gas will remain the dominant heating method for the next decades.
Especially when you mention political (desired) aspects, one can also say that if the state ever needs money, it can quickly impose some kind of tax on gas. Since this is then shared by 50% of households, it will only be a small increase for each and will be approved by the average German without much grumbling.
Basically, almost every (modern) way of heating costs about the same. Almost everyone who has newly built here pays about 800 - 1000 € per year regardless of the heating type, whether gas plus solar, pellets (where you still need space for the pellets) or heat pump. Differences arise mostly through heating or ventilation behavior and the building envelope itself.
Whether one pays 3.80 € less per month with pellets than with gas or 1.20 € more than with a heat pump realistically hardly matters. Every well-calculated and professionally executed heating system is economical nowadays. For comparison, even houses from the 90s, which can still be considered quite new, consume about twice the energy, and here differences of maybe 10, at most 20%, are debated.
For me, gas and heat pump are currently the favorites; if you absolutely want a fireplace, then rather gas; otherwise, you can save the gas connection and also the chimney and rather opt for the heat pump.
As far as aesthetics are concerned, there are also units that can be installed indoors; otherwise just put the outdoor unit somewhere where it does not disturb.