That would probably defeat the entire purpose of Minecraft being on Twitch/Curse in the first place, which is so they have a captive audience to work with (i.e. try to profit off of).
For what it's worth, I totally agree with your question. I don't like how we have to run Curse, then run Minecraft, then close Curse each time. This makes Curse the quintessential "middleman", and I do everything I can do in my life to eliminate middlemen because they are essentially just leeches, sucking off the teat of somebody else's success (in this particular case, Mojang's success).
Don't even get me started on Mojang selling their product to the worst possible corporate entity possible, Microsoft. Just don't even go there. LOL