I understand what you mean. On the other hand, the “quality” of chat messages is usually inferior to forum posts. By converting them totally, you may be creating “chat messages quality” posts and this isn’t necessarily good going forward. It may have an effect of setting the bar lower.
Quoting has the advantage of making very clear the beginning comes from a chat discussion. With the messages form and quality being what’s done in chats. That’s probably good, IMHO.
This would be pretty nice. In addition to the quote, ideally. Or by keeping the quote available, even if masked by default.