A question "why..." in number theory is difficult to answer if it is not just an algebraic but a qualitative one, but let's give a try to improve at least intuition.
For me it is more intuitive to see the (hyper-)operators as (possibly repetatively/iteratively applied) operation at a startvalue, say $x_0$ , having a "base"-parameter $b$ and an iteration-counter $h$.
Then we can also reflect the left- or right-associativity in the completed table:
$\displaystyle \qquad \begin{eqnarray}
G_{1,L}(x_0,b,h) &=& ((((x_0) \underset{\text{h times}}{ \underbrace{+ b) + b) \ldots + b)} }
&\quad= \quad&
G_{1,R}(x_0,b,h) &=& \underset{\text{h times}}{ \underbrace{ (b + \ldots (b + (b +} } (x_0)))) \\
G_{2,L}(x_0,b,h) &=& ((((x_0) \underset{\text{h times}}{ \underbrace{\cdot b) \cdot b) \ldots \cdot b)} }
&\quad =^{1)} \quad&
G_{2,R}(x_0,b,h) &=& \underset{\text{h times}}{ \underbrace{ (b \cdot \ldots (b \cdot (b \cdot} } (x_0)))) \\
G_{3,L}(x_0,b,h) &=&(((( x_0) \underset{\text{h times}}{ \underbrace{ \uparrow b) \uparrow b ) \ldots \uparrow b)} } &\quad \quad & G_{3,R}(x_0,b,h) &=& \underset{\text{h times}}{ \underbrace{ (b \uparrow \ldots (b \uparrow (b \uparrow} } (x_0 ))))
\end{eqnarray}$
$\qquad \qquad ^{1)}$ only over certain (for instance:scalar) fields
Here we have symmetry in the left- or right-associativity $G_{1,L} = G_{1,R} (=G_1)$ and over a scalar field $G_{2,L} = G_{2,R} (=G_2) $
In $G_1$ we have even the interchangeability of the iterator and the base $G_1(x_0,b,h) = G_1(x_0,h,b)$
We can now look at elements with special values in $x_0$ and $b$ to find that the operation is "neutral" ($b=0$ for the addition and $b=1$ for the multiplication and the "power" ($G_{3,L}$)) and becomes "absorbing" for some elements ($b=0$,$x_0=0$ for multiplication and $x_0=1$ for "power" ($G_{3,L}$) and $b=1$ for "exponentiation" ($G_{3,R}$)) and also at the absorbing effects of $\infty$ and of fixpoints in $G_{3,R}$ etc. ...
If I look at all this and even a bit further then the miracle of the "broken symmetry" after the two operations addition and multiplication becomes less mysterious and only one small between other changes of properties (although it is surely remarkable...) .
An additional remark: although I expand here on a hierarchical scheme based on the iteration of operations, I find it worth to remember what Quiaochu Yuan says, that the different operations should be seen in some other view focused at some qualitative(?)/topological(?) properties, for instance the exponential map in the complex plane.