When dealing with multiple sets it is common practice to use convenient notations $\displaystyle\bigcup_{k=1}^nA_k$ and $\displaystyle\bigcap_{k=1}^nA_k$ to mean the union and intersection respectively of sets $A_1, A_2,\dots,A_n$.
I haven't ever seen similar notation applied to the membership ("element of") symbol - i.e. something like $\displaystyle a_k\mathop\in_{k=1}^n A_k$ - though it seems that it would be as convenient and as implicitly understandable as the notation above.
For example, this could allow us to define a Cartesian product very succinctly (and still very understandably) as:
$$A_1\times A_2\times\cdots\times A_n = \left\{(a_1, a_2,\dots,a_n):a_k\mathop\in_{k=1}^n A_k \right\}$$
Or to stretch a point:
$$\mathop\times_{k=1}^n A_k=\left\{(a_1, a_2,\dots,a_n):a_k\mathop\in_{k=1}^n A_k \right\}$$
Is there a reason or convention for the fact that this kind of notation is commonly used for some operators while not others? I can see, for example, that $\cup$ and $\cap$, unlike $\in$ but like $\sum$ and $\prod$, represent operations repeatedly performed on objects of the same kind. Does it have to do with this?
The reason for that is it adds really nothing, your thing can be written as $$\forall i, a_i\in A_i$$ and we are done. And your use of the cross one, we have have $$A_1\times A_2\times\ldots\times A_n = \prod_{i=1}^n A_i$$ to use.
However the main reason some are and some aren't is that those things mark operations of various mathematical objects to produce a new similar object. On the other hand $\in$ doesn't do this. It tells us a relation between two vastely different mathematical objects which is not suitible for such notational development. Another property that is important is that the operation is associative as otherwise it leaves ambiguity and makes it difficult to parse which is intended. For example using the product one within octonions with explicit structure (not just some generic elements of octonions) will be ambigious as it is not certain which order things are multiplied. Even in quaternions it might be troublesome due to non-commutativity.