Which heating system to choose when replacing the heater after 36 years?

Agony of choice,

wouldn't a gas heating system with a tank in the garden and solar thermal make sense?

No. The tank costs money and then you have a tank in the garden. Why so skeptical about the oil?

Oil also requires a tank. Liquid gas is actually an alternative, overall cleaner than oil, but still not completely without all these disadvantages [Tank ...]. But in the present case, one should not ponder too long about it. What is decisive now is the holistic planning, that is the decision whether only the burner/boiler is to be replaced, with everything else remaining, or whether one approaches it holistically with expertise (insulation, windows, energy generator, ...). The latter then goes beyond the scope here and one should consult a professional.

But he does have an intact (?) oil tank...

Yes, there is a welded 4000 l oil tank, according to the previous owner it does not have to be removed. However, I will still clarify with the chimney sweep what all needs to be removed.

We have a similar problem and actually only a new oil heating system makes sense. Currently, oil is cheaper than gas anyway, and I wouldn’t want to bury a gas tank. I also considered a pellet heating system, but I lack the space for a sufficiently large bunker, and it is also unclear how the savings potential will develop over the next few years since wood prices are rising as well. In addition, pellet heating systems are, in my opinion, highly overpriced.

Solar thermal energy is a nice option; I had it installed at my parents’ place. However, the savings are minimal or nonexistent if you have to finance the money for it. Hopefully, we don’t need to talk about heat pumps; they would drive you into ruin.

Regarding your data, something doesn’t add up because your energy consumption seems extremely high to me. I have 30,000 kWh per year, which I find quite reasonable, but I have twice the living space plus a basement. If your bungalow needs 55,000 kWh, then it is really a sieve, and you urgently should consider energetic renovation (that would be 500 kWh/m²a). The previous owner must have lied to you as well because that’s not 1000 liters/year but 3000 liters. I think (hope) your 55,000 kWh is based on 3 years. Then the 1000 liters could fit, and I wouldn’t worry about other heating types. Also, energetic renovation only makes limited financial sense since with your current heating costs of 1600-1700€ per year, it takes a very long time for the corresponding measures to pay off.