I am starting case practices with a group of people this week. During the case practices, one thing I will suggest is to practice selecting the key elements relevant to the problem instead of going through everything in the framework, which clearly shows the interviewer that the interviewee has only memorized the frameworks to attach a problem.
My main problem, I think, was to decide which tree I need to pick first to attach. I had many issues in my mind and was lost in prioritizing them before I proceed. This resulted in a long silence. I find your article about how to avoid getting stuck in case interviews very helpful.
In your practice efforts, I suggest taking as much time as needed to figure out what is important and prioritize. Do not worry about the silence for three reasons.
1) Under the stress of an interview, 10 seconds of silence feels like one minute. In fact, try timing yourself to see how much time you are actually taking.
2) It is much more important to be correct than it is to be fast. If you take a "long" time (or what seems like a long time) up front to gather your thoughts, and you're correct... the interviewer will have completely forgotten about the long opening silence at the beginning of the case.
Especially in practice, take as much as you need to get the setup right. Because if the setup is wrong, the rest of the case will end up being a waste of your time. (It's very difficult, if not impossible, to start a case incorrectly and recover by the end.)
How it starts is how it goes.
3) The more practice you get in prioritizing and assessing what's important, the faster you will eventually get. Focus on effective thinking first, then focus on more time efficient thinking later.