My colleague Dylan answered first (I keep telling him not to spend too much time on this toy :) but I both agree and disagree with his "Yes of course". The same words are used with different meanings and implications in the point-set and infinity category setting. So the word "convenient" has correspondingly different meanings! With the meaning of words in the infty category world, Yes means yes. But the heart of Lewis's argument is that in the point-set world the automorphisms of the unit object of a symmetric monoidal topological category give a point-set level commutative topological monoid. He argues that 1-5 imply that the unit component of $QS^0$ is equivalent to an honest commutative topological monoid, which would imply that it is equivalent to a product of Eilenberg-Moore spaces, which it is not. There is no corresponding contradiction in the infty category world. (Dylan, ok?)
Edit: I'll answer comments in order and add a bit of math to my previous answer. Dylan, enjoy yourself and I'll see you when you get back from Vancouver.
Mike, Gaunce's paper was published in 1991, when Jacob was 13 years old, so long before $\infty$ categories were born. Harry, you have a choice.
The Lewis-May category of spectra satisies 2, 3, part of 4, and 5, but not 1. The EKMM category of S-modules satisfies 1, 3 (but non-cofibrantly, as Dan says), a version of 2, but not 5. I want to expand a bit on 1. In fact, there is a notion of a graded monoidal symmetric monoidal category, never published but known since the 1970's, and the external version of the category of Lewis-May spectra is symmetric monoidal in that sense. The point of EKMM is to
internalize the external smash product while retaining as much as possible of the good connection with spaces. Diagram spectra (symmetric and orthogonal) use a more elementary internalization of a symmetric monoidal graded category, sacrificing the close relationship with spaces. The paper ``Diagram spaces, diagram spectra and spectra of units'', https://msp.org/agt/2013/13-4/p01.xhtml, of John Lind shows how best to relate spectra and spaces starting from different models of symmetric monoidal categories of spectra (symmetric monoidal in the good old-fashioned sense of course :).
The article by Gepner-Groth-Nikolaus is the canonical reference for the tensor product of $E_\infty$-spaces. In the end it is quite a formal construction so there is not that much to say. A useful point of view that does not appear in loc. cit. is that this tensor product comes from the Lawvere theory of commutative monoids. To explain this, consider the $(2,1)$-category $\mathrm{Span}(\mathrm{Fin})$ whose objects are finite sets and whose morphisms are spans $I\leftarrow K\rightarrow J$. It has the following universal property: for any $\infty$-category $\mathcal C$ with finite products, there is an equivalence
$$
\mathrm{CMon}(\mathcal C) = \mathrm{Fun}^\times(\mathrm{Span}(\mathrm{Fin}),\mathcal C),
$$
where $\mathrm{Fun}^\times$ is the $\infty$-category of functors that preserve finite products. Since $\mathrm{Span}(\mathrm{Fin})$ is self-dual, this means that $E_\infty$-spaces are finite-product-preserving presheaves on $\mathrm{Span}(\mathrm{Fin})$:
$$
\mathrm{CMon}(\mathcal S) = \mathcal P_\Sigma(\mathrm{Span}(\mathrm{Fin})).
$$
This was first studied in the thesis of J. Cranch. From this perspective, the direct sum and tensor product are the Day convolutions of $\sqcup$ and $\times$ on $\mathrm{Span}(\mathrm{Fin})$ (here $\times$ means the usual product of finite sets, which is not the categorical product in $\mathrm{Span}(\mathrm{Fin})$; the latter is the same as the categorical coproduct, i.e., the disjoint union $\sqcup$). For example, $E_\infty$-semirings can be described as right-lax symmetric monoidal functors $(\mathrm{Span}(\mathrm{Fin}),\times)\to(\mathcal S,\times)$ that preserve finite products.
The unit. As Rune already explained, the unit for the tensor product of $E_\infty$-spaces is the free $E_\infty$-space on a point, that is the groupoid $\mathrm{Fin}^\simeq$ of finite sets with the $E_\infty$-structure given by disjoint union. This is equivalently the presheaf on $\mathrm{Span}(\mathrm{Fin})$ represented by the point, which is the unit for $\times$ on $\mathrm{Span}(\mathrm{Fin})$.
Here are a few examples I could think of. Let $E\in \mathrm{CMon}(\mathcal S)$.
Tensoring with a free $E_\infty$-space. Let $X\in\mathcal S$. Then
$$
\left(\coprod_{n\geq 0} (X^n)_{h\Sigma_n}\right) \otimes E = \operatorname{colim}_X E,
$$
where the colimit is taken in $\mathrm{CMon}(\mathcal S)$. This follows from the case $X=*$ using that $\otimes$ preserves colimits in each variable.
Tensoring with $\mathbb S$. Tensoring with the sphere spectrum $\mathbb S$ is the same as group-completing:
$$
\mathbb S\otimes E = E^\mathrm{gp}.
$$
For example, for a ring $R$,
$$
\mathbb S\otimes \mathrm{Proj}(R) = K(R).
$$
where $\mathrm{Proj}(R)$ is the groupoid of finitely generated projective $R$-modules, and $K(R)$ is the K-theory space.
Tensoring with $\mathrm{Fin}^\simeq[n^{-1}]$. Another localization of $\mathrm{CMon}(\mathcal S)$ is obtained by inverting integers (or rather, finite sets). The inclusion of the full subcategory of $E_\infty$-spaces on which multiplication by $n$ is invertible has a left adjoint $E\mapsto E[n^{-1}]$, which is equivalent to tensoring with $\mathrm{Fin}^\simeq[n^{-1}]$. But unlike in the cases of either abelian monoids or spectra, $\mathrm{Fin}^\simeq[n^{-1}]$ is not just the sequential colimit of multiplication by $n$ maps; it is obtained from the latter by killing suitable perfect subgroups of its fundamental groups, in the sense of Quillen's plus construction, to ensure that $n$ acts invertibly.
Tensoring with $\mathbb N$. Let $\mathrm{FFree}_{\mathbb N}$ be the 1-category of finite free $\mathbb N$-modules. There is a functor
$$
\mathrm{Span}(\mathrm{Fin}) \to \mathrm{FFree}_{\mathbb N}
$$
sending a finite set $I$ to $\mathbb N^I$, inducing an adjunction
$$
\mathrm{CMon}(\mathcal S) = \mathcal P_\Sigma(\mathrm{Span}(\mathrm{Fin})) \stackrel{\mathrm{str}}\rightleftarrows \mathcal P_\Sigma(\mathrm{FFree}_{\mathbb N}).
$$
Objects in the right-hand side are sometimes called strictly commutative monoids (the group-complete ones are connective $H\mathbb Z$-module spectra). Tensoring with $\mathbb N$ amounts to strictifying a commutative monoid in this sense:
$$
\mathbb N\otimes E = E^\mathrm{str}.
$$
Unlike $\mathbb S$, $\mathbb N$ is not an idempotent semiring, that is, strictifying is not a localization. Indeed, $\mathbb N\otimes\mathbb N$ is an $E_\infty$-space whose group completion is the "integral dual Steenrod algebra".
Tensoring with $\mathrm{Vect}_\mathbb{C}^\simeq$. Let $\mathrm{Vect}_\mathbb{C}^\simeq=\coprod_{n\geq 0} BU(n)$, where $U(n)$ is regarded as an $\infty$-group (despite the notation, this is not really the core of an $\infty$-category of vector spaces). This is an $E_\infty$-space whose group completion is $\mathrm{ku}$. There is a related $\infty$-category $2\mathrm{Vect}_{\mathbb C}$ whose objects are finite sets and whose morphisms are matrices of complex vector spaces. As in the previous example we get an adjunction
$$
\mathrm{CMon}(\mathcal S) = \mathcal P_\Sigma(\mathrm{Span}(\mathrm{Fin})) \rightleftarrows \mathcal P_\Sigma(2\mathrm{Vect}_{\mathbb C}).
$$
An object in the right-hand side is roughly speaking a commutative monoid such that $U(n)$ acts on the multiplication by $n$ map in a coherent way. Tensoring with $\mathrm{Vect}_\mathbb{C}^\simeq$ gives the free commutative monoid with such structure.
Best Answer
For $E_\infty$ spaces, homotopy-theoretically there is a functor $L: \mathcal{S} \to E_\infty \mathcal{S}$ with a right adjoint $R$. The only property on this list that really needs replacing on this list is property (5): the unit $$ X \to RL(X) $$ should be homotopy equivalent to the natural inclusion $$ X \to Free_{E_\infty}(X) \simeq \coprod_{k \geq 0} E \Sigma_k \times_{\Sigma_k} (X^k) $$ into the free $E_\infty$-space on $X$. (Yes, yes, possibly a version with basepoints, I know)
I believe that all three of the models of $E_\infty$ spaces that you list (commutative monoids in $*$-modules, $\Gamma$-spaces, commutative $\mathcal{I}$-space monoids) satisfy properties (1)-(4) and fail the analogue of property (5), each due to an issue about whether an input to an adjunction is cofibrant/fibrant. For $\Gamma$-spaces, for example, the map $X \to RL(X)$ only adds a disjoint basepoint. Perhaps someone with more experience with the other models would be able to fill in those stories better.