Yes, mine started several days ago as well. I often get the captcha screen on Youtube when surfing on my computer. However obviously having a keyboard the problem is solved. The other day when Chromecast started showing the screen there was nothing I could do. I tried to stream from my tablet and my phone, but neither showed the captcha screen. Only the TV showed it. This is because Chromecast is seen as it's own device, like a mini computer.
I have been emailing with Google and called the phone number to discuss. What's been more worrisome is that so far I don't feel confident that they understand what I'm talking about. Which seems crazy. My next plan is the next time it does it bad like it did the other day where I had to give up I plan to video it and post on YouTube to send them a link. It's the only thing I can think of for them to get the message.
I also did what was suggested by going to another computer and getting the screen to come up on it's own hoping that if I entered the words it would work. Nope no luck. The captcha wants to know that the particular device is being maned by a human and not just an autonomous device trying to get YouTube hits. So Chromecast is on it's own unless they do a software patch to fix them all to skip the captcha screen.
We shall see