It could work but as I said it would be dishonest. If a bot can read your messages then you have to trust a third party to not leak/store them. I believe that he goal of E2E encryption is to remove the need to trust somebody. (Of course you still have to trust FB that they do not backdoor or hinder the encryption but you do not have to trust the third parties)
I do not speak of chatbots such as Cleverbot. For example your bot will listen to your messages and provide contextual information, such as theatres airing a movie you are currently talking about. The bot also probably needs to keep track of the context, which probably means keeping the chat history for some time. In order to learn the chatbot needs to keep the history forever and scan it for ML purposes. All of these mean that the bot has to keep the history somewhere unencrypted, basically nullifying the benefit of E2E.
It might be true that FB adding E2E breaks existing bots, but it's false that it breaks that ability to use bots on FB.