Is it unreasonable to expect someone you pay $75 - $120 and hour to to know the syllabus without looking?

In my opinion I should be able to point to any figure in the ISTD syllabus and the "pro" should be able to immediately be able to demonstrate the proper execution of that figure.

I think there is a difference between being able to describe the figure verbatim from the book and being able to demonstrate it accurately.

For the most part, a pro who is teaching advanced students would be able to demonstrate it accurately - although on occasion I don't think it is a bad thing that they refer to the book to refresh their memory about whether a certain step is a heel, or up or down, or whatever. Especially with a figure that is not used as often or has more than one variation. (and when it's the steps of the opposite gender they're confirming)

Who in any of their professions can remember EVERYTHING they're supposed to know? Doctors, lawyers, teachers... To me there is a difference between temporarily forgetting something and needing to check back - and just plain not knowing. A pro is someone who can properly interpret the information and then deliver it accurately - that's why you pay big money...not because they're a machine that can just parrot back the explanations.

I can't say this applies to everyone's pros - but mine are quite successful in their career and every once in a while we stop to double check something in the book. And it's never been cause for me to put their skills in doubt.

The qualifying examinations of the main teaching societies are meant establish that the technique which is appropriate to the syllabus figures has been committed to memory AND that the figures can be demonstrated, as man and lady, to a professional standard.

Anyone who teaches day in, day out, will find the process of recall gets easier, with experience; and so, yes, a properly trained and qualified teacher should be able to recall and demonstrate anything in the book; but don't forget that an Associate needs to know rather less than a Fellow, and may only be qualified in one branch, such as Ballroom, but not Latin.

Just how much time teachers spend, privately, refreshing their knowledge, is their own business, but I don't expect to pay good money for a lesson, to have the teacher needing to look up the answer to my question.

I absolutely agree! And you must read your own situation and how often it is happening. I'm just saying that for me, when it happens every once in a blue moon, I don't immediately judge their quality of knowledge. People make mistakes and forget things from time to time. (Doing it ALL the time, or frequently, is unacceptable.)

However, anyone who has studied for the exams knows that knowing the book isn't enough to teach well. So much of what makes a good dancer can't be written down.
As well, when teaching you often don't provide all the information all the time (depending on the skill level of your student it would often be over their head!) So yes, the info you repeat will be solid, but you don't use all the info in the book everyday in your teaching - that's why it's called double checking or refreshing your memory. Not reading up to learn how!

I guess I'd be quite impressed if I ever met anyone who could remember everything all the time...

Set your expectations high - but remember you're still working with people.