If the candidate would like some quiet time to think about the solution, I expect them to respond with "I think I'm forming a solution, just give me a couple of minutes to think about it before I present my case".

If they can't even do that, then I'm sorry, but they are no good to anyone. You can be the smartest and best developer in the world, but if you're completely incapable of representing yourself and your ideas; to have a dialog about your work, you are effectively worthless as an employee.

Some companies might have a place for that special someone, who you can lock in a room for a month and he will later emerge with an amazing new piece of code that will solve your problems, shielded from all the problems of the outside world, but companies where that is possible are very rare.

Also, being able to develop like that is very rare because in the real world, requirements change or are vague and need to be clarified, or your software has to interoperate or integrate with other software. There is simply no way to avoid regular communication and still produce something that works how it needs to. Problems where you can shut yourself away for a long period of time certainly exist, but in the grand scheme of software jobs, they're few and far between.

The point is that with this kind of make or break style of candidate vetting you're probably leaving a lot of great talent by the wayside.

The whole point is to make them comfortable and that takes the highest priority. Being able to read the individual and react to them is key, not everyone is the same, hence why there are a multitude of responses, and the ones given are just a starting point.

I.e. if stuck at something, even if seeming trivial, it might pay (to go back or have a chat around it, even guide him/her a bit so as, etc) to get the candidate loosened up and feeling safe which should then yield better responses after.

There is no one-size-fits-all interview technique; if there was, it would be used all over the place. Good interviewing techniques are about playing the numbers, not leaving no programmer behind.