One should distinguish at least between adnominal, prepositional and adverbal Genitive here.
Genitive case as an indicator of possession is very common. This is a usage you know from English. For example,
Peters Freund hat Ingrids Buch genommen.
Wessen Buch hat er genommen?
Wessen Buch wurde genommen?
All of these sentences are perfectly acceptable and stylistically fine.
There are some alternatives, but Genitive is neither "weird sounding" nor wordy. Neither of the following options is strictly preferable:
Das Buch von Ingrid wurde gegessen.
Dem Peter sein Freund war das.
This is less clear for cases other than a person's name, e.g.
Frankfurts Straßen sind schön.
Die Straßen von Frankfurt sind schön.
Here, the prepositional variant may be a bit better, but the Genitive variant is still okay. But it quickly becomes impossible or very stilted for longer possessors:
Des Supermarkts Preise sind okay.
This is an awkward-sounding sentence. However, here, the best option is a post-nominal Genitive possessor:
Die Preise des Supermarkts sind okay.
but also (without Genitive)
Die Preise vom Supermarkt sind okay.
So here, the non-Genitive options are often (but not always) equally good or preferable. In dialects, the alternative forms of indicating a relationship are sometimes better. Nonetheless, this adnominal, possessive Genitive is very much alive and well.
Prepositional Genitive is in many instances replaced by the Dative, e.g. the semi-famous example used by a much-deried and ridiculed German prescriptivist:
Wegen des Umbaus geschlossen.
Wegen dem Umbau geschlossen.
However, there are some instances that actually used to be Dative prepositions and now show the Genitive, e.g. trotz.
Finally, adverbal Genitive has become very rare and usually sounds a bit stilted, e.g.
Er bezichtigte ihn des Verbrechens.
In most cases, "higher"/more formal registers of German show more propensity towards using the Genitive, and "lower"/informal registers as well as most dialects often use alternatives. In summary, adverbal Genitive is very rare and you usually won't encounter it. However, at least for the adnominal Genitive and many instances of prepositional Genitive, it is very common.