I wonder if we can construct some different geometric realizations?
In the current construction, $n$-cells of geometric realization are in one-to-one correspondence with non-degenerate simplices in $X_n$, which is due to the equivalence relation given by $$(f^*(x), t)\sim (x, f_*(t)), x\in X_n, t\in \Delta^m$$ for every morphism $f : [m]\to [n]$ in $\Delta$
(Recall the geometric realization is defined by $|X_*|= (\bigsqcup_{n\ge0} X_n\times \Delta^n)/\sim$).
I think if we drop the equivalence relation, we can get a new CW-complex, where $n$-cells of construction are in one-to-one correspondence with simplices in $X_n$.(Am I right? I am not very sure, but it seems that works)
Also, is there a specific reason/motivation for the current construction of the geometric realization? Why do we require $n$-cells of geometric realization are in one-to-one correspondence with non-degenerate simplices in $X_n$?
The geometric realization of a simplicial set is only unique up to (topological) isomorphism. There is no overriding reason to define as you do above. The definition of geometric realization you give is chosen firstly so that
1) It is functorial in simplicial maps $X_\bullet\rightarrow Y_\bullet$.
2) It defines a left adjoint to the singular set functor $Top\xrightarrow{S}sSet$ definedby $SY_n=Map(\Delta^n,Y)$.
and also so that
3) For 'nice' classes of simplicial sets it will take values in 'nice' classes of topological spaces.
In particular the geometric realisation $|-|$ depends on the choice of the standard topological simplices $\Delta^n$, the definition of the singular set functor $S$ and how we want it to interact with certain classes of simplicial sets and spaces. It is "defined" as above by many texts simply because it is the most convenient possible definition.
Indeed a similar definition is that of the fat geometric realization, given by
$||X_\bullet||=\bigsqcup_{n\geq 0} X_n\times \Delta^n /\left[(d_ix,t)\sim (x,d^it)\right]$
In particular the fat geometric realization does not throw away the degenerate simplices. There is a natural map
$\alpha:||X_\bullet||\rightarrow |X_\bullet|$,
and if $X_\bullet$ is good, that is, if each degeneracy map $s_i:X_{n-1}\rightarrow X_n$ for each $n$ is a closed cofibration, then $\alpha$ is a homotopy equivalence. It has other nice properties. If $f_\bullet:X_\bullet\rightarrow Y_\bullet$ is a levelwise homotopy equivalence then $||f_\bullet||$ is a homotopy equivalence whilse $|f|$ need not be. See for example Segal's paper 'Categories and Cohomology theories'.