What other alternatives might there be to using a public exchange? How can we turn media apps - like
Telegram - into private Bitcoin transfer places that might 'push' Bitcoin into private use among a lot more people?

Unfortunately, with the huge number of scam accusations in media apps especially in Telegram, people think it was not a safe way of buying/selling Bitcoin. They rather prefer to use centralized platforms even if it compromises their privacy. Some people think that the use of government-controlled exchanges is safer than the rest.
Everything went to private transfers if there were no scammers taking advantage of it but sadly, they are ruining this. I'm not sure if this is a sort of misunderstanding because what I see is that they just want safety assurance in all of their in/out transactions.