In the Hott book, are the most of the type formers redundant? And if so, why?
You are asking several questions which are similar but distinct. 


Why doesn't the HoTT book use Church encodings for data types?

Church encodings do not work in Martin-Löf type theory, for two reasons. 

First, MLTT is predicative. There is a universe hierarchy, and each type lives at a particular universe level, and a type at level $n$ can only quantify over types at smaller universe levels $k &lt; n$. Church encodings like in System F or the CoC require impredicative quantification, where you can instantiate quantifiers with types which are the same size, or even bigger. 

Second, even if you defined datatypes like the natural numbers with Church encodings, to do proofs with these types, you need induction principles to prove things about them. To derive induction principles for Church encodings, you need to use an argument based on Reynolds' parametricity, and the question of how to internalize parametricity principles into type theory is still not fully settled. (The state of the art is Nuyts, Vezzosi, and Devriese's ICFP 2017 paper Parametric Quantifiers for Dependent Type Theory -- note that this is well after the HoTT book was written!) 
Next, you are asking why the foundation is not minimal. This is actually one of the distinctive sociological features of type-theoretic foundations -- type theorists regard having a small set of rules as a technical convenience, without much foundational significance. It's far, far more important to have the right set of rules, rather than the smallest set of rules. 

We develop type theories to be used by mathematicians and programmers, and it's very, very important that the proofs done within type theory are the ones that mathematicians and programmers regard as being done in the right way. This is because the arguments mathematicians typically regard as having good style typically are structured using the key algebraic and geometrical principles of the domain of study. If you have to use complicated encodings then much structure is lost or obscured. 

This is why even type-theoretic presentations of propositional classical logic invariably give all the logical connectives, even though it is formally equivalent to a logic with just NAND. Sure, all the boolean connectives can be encoded with NAND, but that encoding obscures the structure of the logic.