Here is a way to write down what a symmetric monoidal category is starting just from the idea of finite (co)products. We have
$$SymMonCat = Fun^\times_{(2,1)}(Span^{fin}_{(2,1)}, Cat_{(2,1)})$$
That is, a symmetric monoidal category is equivalent to the data of a finite-product-preserving $(2,1)$-functor from the $(2,1)$-category $Span^{fin}_{(2,1)}$ of finite sets and spans between them, to the $(2,1)$-category $Cat_{(2,1)}$ of categories.

I can think of at least two ways to understand the significance of $Span^{fin}_{(2,1)}$ from the point of view of finite (co)products.

- There is a functor $Span^{fin}_{(2,1)} \to Cat^{\amalg}_{(2,1)}$ (where $Cat^{\amalg}_{(2,1)}$ is the $(2,1)$-category of categories with finite coproducts and finite-coproduct preserving functors). The functor sends $n \mapsto Set^{fin}/n$. This functor is fully faithful. Moreover, $Set^{fin}/n$ is itself the free coproduct completion of the discrete category $n$.

So from this perspective, we have boiled it all down to finite (co)products, but the picture is a bit odd and I'm not sure how to complete the thought from this perspective. Alternatively,

- Note that the $(2,2)$-category $Span^{fin}_{(2,2)}$ of finite sets, spans between them, and maps of spans has the following universal property. For any 2-category $\mathcal K$ with finite products, we have that the 2-category $Fun_{(2,2)}^\times(Span^{fin}_{(2,2)}, \mathcal K)$ of all finite-product preserving 2-functors, is equivalent to $FinProd(\mathcal K)$, the 2-category of objects of $\mathcal K$ which internally have finite products.

(2) suggests the following generalization. Let $J$ be a set of small categories, and let $2Cat(J)$ be the category of 2-categories with $J$-limits. I believe that the functor $Comp_J: 2Cat(J) \to 2Cat$ carrying $\mathcal K$ to the 2-category of objects in $\mathcal K$ which internally have $J$-limits, is corepresentable by some $\mathcal J \in 2Cat(J)$. Define a *$J$-symmetric monoidal category* to be an object of $Fun^J_{(2,1)}(\mathcal J_{(2,1)}, Cat)$, where $\mathcal J_{(2,1)}$ is the maximal sub $(2,1)$-category of the $(2,2)$-category $\mathcal J$, and $Fun^J_{(2,1)}$ means we take $(2,1)$-functors which preserve $J$-limits in the $(2,1)$-categorical sense.

Then in the case where $J$ consists of the finite discrete categories, a $J$-symmetric monoidal category is a usual symmetric monoidal category. When $J$ consists of just the empty category, a $J$-symmetric monoidal category is an $E_0$-category. You could try out other classes of $J$, such as equalizers, but I'm not sure what you get!