Eugene,
I experienced a similar thing when trying my authentic voice in regular conversation; I’d get maybe 1/6 of my thoughts out before being interrupted (not necessarily in a bad way) or getting self-conscious and stopping early. That said, it was fun to use an initial piece of my inner monologue to begin a conversation. I’d start speaking whatever was on my mind and allow others to chime in. It wasn’t quite the same as the assigned exercise but it definitely led to good conversation and connection that felt meaningful and more authentic.
So while the full exercise may not be suited very well to regular conversation, I think it can be practiced in bits throughout a regular day.