Download an image from Facebook into browser's private downloads directory, copy it using a file-manager application (one of the exceptional applications having full filesystem access) into Telegram's private directory, upload it into chats you need to post it.

The file-manger application managed above is a single point of failure, of course. So, it should be allowed to use only one provided by OS vendor.

Sounds like effort, people don't like effort when spreading memes, so would be enraged if this would be the default now, or nobody would activate it.

There is a lot of middle ground, like having a shared folder for access. "Downloads" might be a good default, if clearly communicated, that anything in there, is accessible by any app.