How I find a house that suits me within an appropriate timeframe

I think it's best if you hire a nationwide real estate agency, share your wishes with them, and then they should deliver.

Since you are flexible: I am currently looking for new neighbors.

Next door to me, a nice little house is for sale. Less than 2 years old. Address available by PM.

I don’t want to live here anymore right now (I live in NRW), so I am flexible nationwide.

Then make yourself a plan about what you want. Your search is totally unstructured and eBay Classifieds is the worst possible platform to look for a property. Do you have a job? Which location is important – rather the North Sea coast or the Alpine foothills? Airports nearby already narrow down the choice significantly; what do you need the airport for? If you know the region that suits you, you can search more concretely. Although the metropolitan areas almost fall out, since your budget might be too small (unless an airport nearby means an hour’s drive. Then you can buy almost anywhere in Germany).

and airport is not just any airport. The possibilities of FRA are naturally different from, for example, Kassel-Calden, or to stay with Frankfurt, the airport Frankfurt-Hahn.

Nice article (18.08.) on the current airport situation and accessibility of airports:
As a result of the corona crisis, several subsidized regional airports in Germany could be on the brink of bankruptcy, according to a newspaper report. In the short term, the existence of 7 of the 14 regional airports examined is threatened, warns a study that the Forum Ecological-Social Market Economy prepared for the environmental organization BUND and which is available to the "Süddeutsche Zeitung" (Wednesday).
In the medium term, even twelve of the airports that depend on substantial state aid are at risk of closure. The study rates the prospects for the airports Frankfurt-Hahn, Kassel-Calden, Niederrhein-Weeze, Paderborn/Lippstadt, Rostock-Laage, Erfurt-Weimar, and Saarbrücken as particularly critical. The study will be presented publicly on Wednesday.
The corona crisis is only part of the problem: With 24 major commercial airports, including the regional airports, massive overcapacities have been built up in air traffic in Germany, the paper continues. The large international airports in the country are at most 100 train minutes away from the smaller locations.

Even with us, it only works through connections, word of mouth. What doesn't get sold there goes on the market.

I would narrow down the region and then search there, not through Ebay Kleinanzeigen.