Photovoltaic system: Advantages and disadvantages, experiences, worthwhile?

I am a beginner in this field and I still confuse the technical terms. What I meant by economical is more about how much electricity can be saved with a photovoltaic system since less has to be drawn from the supplier. And after how many years it pays off compared to the purchase price. On the topic of battery: This would serve as a buffer, for example, when no one is home during the day and only the basic consumers (heat pump, ventilation system, devices in standby) are running, but the photovoltaic system generates more electricity than is needed. This way, electricity can be provided for the evening hours when demand rises again, thus reducing the amount of purchased electricity? Sorry if this sounds very amateurish, I lack experience - but maybe you can follow my idea. Primarily, photovoltaic and battery are supposed to better capture self-consumption and make one more independent from peak times (sunshine hours).

Primarily, photovoltaics and batteries are intended to better cover self-consumption and to make oneself more independent from peak times (sunlight hours).

Well, the battery also has to be paid for well. In addition, self-consumption is not free either and must be taxed. Think carefully about how much electricity you really need at night during sunny periods (when it is light for a long time in the evening). Which devices should be powered?

I currently find batteries still too expensive to use them economically.

I will try to explain it simply again.

You install a photovoltaic system on your roof; it produces a calculable amount of electricity per year.
In a normal household, you can directly consume about 30% of it yourself and feed 70% into the grid.
These 30% can vary for better or worse, so optimally, for example, you would use a heat pump to prepare hot water during the day, run the washing machine and dryer, or as you write, cool with the heat pump.
You get paid for the 70%.
Depending on the system size, you will then be paid more or less than what you have to pay for the rest of the electricity you buy.
Either way, after about 10-12 years, you have made back the costs for the system through this principle. Dispassionately viewed, it’s simply an investment with about 3-4% return per year, which is quite good given the current zero interest rate policy.

If you want to consume as much electricity as possible yourself, i.e. more than 30%, that would be possible with a battery. You can size it so that you hardly have to buy electricity from April to October. That sounds great at first, but the problem is that a battery costs as much as you would pay for this electricity over 20-30 years, given the fact that the battery lasts at best 15 years – so it is a losing proposition.
If money is not very important to you and it’s just about using a lot of self-produced electricity, it is a nice thing.

I currently find batteries still too expensive to use them really economically.

On the other hand, you bring a fire risk into your home. Although it is in the PPM range, if that thing catches fire, you can say goodbye to your house. There's no putting it out...

I would buy my system again and again. I generate 50% of the total electricity required myself, the rest goes away... the system has already paid for itself after 7 years. I am very satisfied.

Really 7 years? Mine had already paid off after 1 year. That took quite a long time for you..