The answer to your question is known when $\mathcal{S}$ is a symmetric monoidal $\infty$-groupoid, by work of Galatius-Madsen-Tillmann-Weiss.
In other words: we understand invertible $2$-dimensional TFTs.
This might seem like a somewhat silly case, but it's actually a very useful way of testing conjectures.
Below I'll sketch how this implies that the answer to Noah's question is no, but let me begin by explaining the setup.

Let's a assume that every morphism in the symmetric monoidal $(\infty,1)$-category $\mathcal{S}$ is invertible and that every object of $\mathcal{S}$ is invertible under $\otimes$.
(The latter could be replaced by the assumption that every object is isomorphic to the unit $1_{\mathcal{S}}$ - it doesn't really matter since $B(\mathrm{Cob}(2))$ is connected.)

A symmetric monoidal $\infty$-groupoid is an $E_\infty$-algebra in spaces, and with the above invertiblity assumption it is an infinite loop space $\mathcal{S} = \Omega^\infty X$ of some connective spectrum $X$.
Now every functor $\mathrm{Cob}(2) \to \mathcal{S}$ factors through the groupoidification
$\mathrm{Cob}(2) \to (\mathrm{Cob}(2))^{\rm gp}$, which can be modeled as the classifying space of the $\infty$-category $\mathrm{Cob}(2)$.
This classifying space was computed by Galatius-Madsen-Tillmann-Weiss as:
$$B(\mathrm{Cob}(2)) \simeq \Omega^\infty( \Sigma^{-1} \mathrm{MTSO}_2)$$
Here $\mathrm{MTSO}_2$ is a certain Thom spectrum, which can be studied by standard methods in stable homotopy theory. (I'm happy to elaborate, if you like.) Anyways, putting this together we have that:
$$
\mathrm{Fun}_\infty^\otimes(\mathrm{Cob}(2), \mathcal{S})
\simeq \mathrm{Map}_{\mathrm{Sp}}(\tau_{\ge 0} \Sigma^{-1} \mathrm{MTSO}_2, X).
$$
Here $\tau_{\ge0}$ is the functor that sends a spectrum to it's connective cover.
~~This could be dropped since $X$ is connective, but is seems more accurate to keep it around as $\mathrm{Cob}(2)$ doesn't ~~*know* about $\pi_{-1} \Sigma^{-1} \mathrm{MTSO}_2 = \mathbb{Z}$.

So why is this useful at all? I'll try to illustrate that be giving a negative answer to the question Noah posed in his post.
I'll say *traced* $E_2^{\rm fr}$-*algebra* to mean a framed $E_2$-algebra $A$ equipped with an $SO_2$-equivariant trace $\tau:A \to 1$.
Let $\mathcal{C}$ be the free symmetric monoidal $(\infty,1)$-category on a traced $E_2^{\rm fr}$-algebra.
Since $S^1 \in \mathrm{Cob}(2)$ has this structure, there is a symmetric monoidal functor $F:\mathcal{C} \to \mathrm{Cob}(2)$ sending $A$ to $S^1$.
Noah's question can now be reformulated as asking whether for all symmetric monoidal $(\infty,1)$-category $\mathcal{S}$ precomposition with $F$ induces a fully faithful functor
$$
\mathrm{Fun}_\infty^\otimes(\mathrm{Cob}(2), \mathcal{S})
\longrightarrow
\mathrm{Fun}_\infty^\otimes(\mathcal{C}, \mathcal{S}),
\qquad \mathcal{Z} \mapsto \mathcal{Z} \circ F
$$
whose image consists of those $\mathcal{Z}': \mathcal{C} \to \mathcal{S}$
such that $\mathcal{Z}'(A \otimes A \xrightarrow{ \mu } A \xrightarrow{ \tau } 1_{\mathcal{C}})$
is a non-degenerate pairing.

If $\mathcal{S}$ is an $\infty$-groupoid, then this condition is trivially satisfied, so we should have an equivalence between the functor categories.
This implies that $F$ induces an equivalence $B\mathcal{C} \to B\mathrm{Cob}(2)$.
However, one can show that this is not true. I'll be very brief on this.
Any $E_2^{\rm fr}$-algebra in an $\infty$-groupoid can canonically be trivialised along the unit morphism $1_{\mathcal{S}} \to A$,
so we basically just have to give an $\mathrm{SO}_2$-equivariant morphism $1_{\mathcal{S}} \to 1_{\mathcal{S}}$ where both sides are equipped with the trivial action.
This implies that:
$$
B(\mathcal{C}) \simeq \Omega^\infty \Sigma^{\infty+1} (B\mathrm{SO}_2)_+.
$$
So the answer to the question is no since the connective spectra
$\Sigma^{\infty+1} (B\mathrm{SO}_2)_+$ and $\tau_{\ge 0} \mathrm{MTSO}_2$
are not equivalent.

The above argument is very roundabout and actually quite subtle:
there is actually a map of spectra
$$
\tau_{\ge 0} \mathrm{MTSO}_2 \to \Sigma^{\infty+1} (B\mathrm{SO}_2)_+
$$
the fiber of which is $\tau_{\ge0} \Sigma^{-1}\mathbb{S}$.
(This is known as the Genauer fiber sequence and admits an interpretation on bordism categories.)
So in particular this map is a rational equivalence.
This might make us hopeful as it looks like we weren't that far off.
However, this map goes the wrong way and I actually believe that there is no rational equivalence that goes the correct way around.

So let me try to give a more down-to-earth proof that the functor $F:\mathcal{C} \to \mathrm{Cob}(2)$ is not an equivalence on groupoidification.
For a traced $E_2^{\rm fr}$-algebra $A$ in $\mathcal{S}$ one can define $\tau(1):1_{\mathcal{S}} \to A \to 1_{\mathcal{S}}$ as the composite of unit and trace.
If the traced $E_2^{\rm fr}$-algebra comes from a $2$-dimesional TFT, then this is the value on the $2$-sphere.
The $\mathrm{SO}_2$-action on $A$ gives us a $2$-morphism $\alpha:\tau(1) \Rightarrow \tau(1)$. (The Dehn-twist pre- and post-composed with cap and cup.)
Since both $1:1_{\mathcal{S}} \to A$ and $\tau:A \to 1_{\mathcal{S}}$ are $\mathrm{SO}_2$-equivariant we have two trivialisations of $\alpha$.
Together they yield a $3$-morphism $\gamma: \mathrm{id}_{\tau(1)} \Rrightarrow \alpha \Rrightarrow \mathrm{id}_{\tau(1)}$.
In the above I didn't actually use that $\mathcal{S}$ is an $\infty$-groupoid;
one can construct this $\gamma$-invariant for any traced $E_2^{\rm fr}$-algebra in any $(\infty,1)$-category.

If the traced $E_2^{\rm fr}$-algebra comes from a $2$-dimensional TFT $\mathcal{Z}:\mathrm{Cob}(2) \to \mathcal{S}$, then $\tau(1) = \mathcal{Z}(S^2)$ and $\alpha$ is the Dehn twist along the equator.
The $3$-morphism $\gamma: \mathrm{id}_{\mathcal{Z}(S^2)} \Rrightarrow \mathrm{id}_{\mathcal{Z}(S^2)}$
probably corresponds to the $S^1$-family of rotations of $S^2$ around some axis.
In particular this $3$-morphism is always of order $2$ since $\pi_1 \mathrm{Diff}^+(S^2) = \pi_1 \mathrm{SO}_3 = \mathbb{Z}/2$.

Ok, so now all that's left to show is that there are traced $E_2^{\rm fr}$-algebras where $\gamma$ is not of order $2$.
For this purpose let $\mathcal{S}$ be $K(\mathbb{Z},3)$ thought of as a symmetric monoidal $3$-groupoid that has a single object, morphism, $2$-morphism, and $\mathbb{Z}$-many $3$-morphisms.
Then we can take the trivial $E_2^{\rm fr}$-algebra $1_{\mathcal{S}}$ in there,
equip it with the trivial trace $\tau:1_{\mathcal{S}} \to 1_{\mathcal{S}}$,
and give this trace non-trivial coherence data.

This ended up much longer than intended, but I hope it's helpful. Let me know if I should clarify anything.

I'd be very curious to see a conjectural generators and relations description of $\mathrm{Cob}(2)$ that yields both the right homotopy category and the right classifying space.
(Though this seems to be very difficult as the $\Omega^{\infty} \Sigma^{-1} \mathbb{S}$ that turned up above is in some sense not finitely generated...)

classicalfield theories. $\endgroup$