You are correct in that there are many ways of counting assumptions. For example, the assumption is made that the world has a history prior to the present moment which is somewhat consistent with our memory and our perception of historical evidence. And so on. It was in that spirit of completeness that I included the English language as one tool Dick used, which of course, contains a wealth of assumptions about the meaning of each word used.
It would be rather silly to try to itemize all the assumptions at that level before starting on an enterprise such as Dick's. Instead, Dick cited mathematics as being his foundation, which I think we can agree would include the assumptions relating to the English language and the Peano Axioms. That was what I meant by using the term 'tools'.
Beyond the tools, Dick clearly states the assumption that any universe which is to be considered as the subject of his analysis must be communicable, i.e. any features of interest in that universe must be expressible as numbers. I am sure Dick would not deny making that assumption.
(BTW there has been a lot of unfortunate misunderstanding caused by Dick's choice and use of the words 'universe' and 'reality'. In my view, these are merely symbolic tokens for the set of numbers which is the actual subject of his work. The token 'universe' is commonly used in the mathematical field of Statistical Analysis to refer to the entire population under consideration, and Dick's usage is consistent with that. I remain convinced that all the difficulties surrounding the choice of those terms are fluff that get in the way of seeing that Dick's result is really a theorem that belongs in Statistical Analysis. But, alas, I seem to be the only person so far who sees things that way.)
Now, as for shift symmetry, I doubt that I can clear up that confusion for you. Dick does not assume shift symmetry. He makes no assumptions about, and sets no conditions on, his set of numbers. Shift symmetry falls out as a consequence of that arbitrariness, thus it is not an assumption.
I hope that helps. It's the best I can do.