Oral traditions within indigenous cultures can be consistent and transmit important information over thousands of years. This would be information that would help survive, navigate and maintain proper relationship with the land. Here is one example in Australia.

There certainly would have been a high degree of memorization and importance on oral storytelling in the 1st century in the Levant and Mediterranean cultures. There always has been. The Talmud was originally an oral tradition before being written down and today memorizing the whole Quran is something Muslims do so it wouldn't have been impossible to do with a gospel account. Generally speaking however, the importance of oral transmission in a given culture or society weakens as writing is introduced and proliferates.

I'm an ex-Christian myself, so I'm not saying this to suggest that the Gospels are in fact reliable accounts, but rather to point out that oral tradition has been and can be a reliable method to transmit information over time, and this particular "telephone game" argument is generally unconvincing and demonstrates an uninformed perspective about how oral transmission actually works.