In Goodwillie's Calculus I he defines an excisive functor to be one, which maps homotopy pushouts to homotopy pullbacks.
Why the change of universal properties?
It is surprisingly hard to find excision or Mayer-Vietoris theorems phrased in terms of homotopy pushouts. Failing to do so, they seem to be mostly framed as preserving homotopy pushouts, for example Kerodon 3.4.6.1 or Eisenbud-Harris 1.14. What I have in mind is that a generalized homology theory should essentially be a covariant functor, which preserves homotopy colimits. (Surprisingly Tamme talks about excision in the context of preserving homotopy pullbacks, but I digress)
So if we want to have an excisive approximation for functors $\mathcal{C} \rightarrow \mathcal{D}$ between general homotopy theories, it feels more natural to require the preservation of homotopy pushouts. This is acknowledged in remark 1.5 of Arone-Ching's survey of Goodwillie calculus, but they only offer the explanation that the change of universal properties makes the calculus more useful / interesting. I feel this is somewhat unsatisfying (if I would tell someone doing calculus, maybe it would become more interesting if the derivative of the identity wouldn't be trivial, I don't want to know what will happen...).
Another a posteriori reason for the change of UPs is of cause, that $\Omega^\infty\Sigma^\infty$ exhibits this property, and if we consider functors $\mathcal{C}\rightarrow\mathcal{D}$ with $\mathcal{D}$ stable, there is no distinction. But this incorporates into the motivation, that 1) the derivative of ${\operatorname{id}}_{\mathsf{Top}}$ is nontrivial and 2) that it is given by $\Omega^\infty\Sigma^\infty$. I find this unsatisfying as well.
Phrased differently:
Am I wrong that excision ought to mean preserving homotopy pushouts? And if not, why is Goodwillie's definition still the right one?
No, excisive shouldn't mean preserving homotopy pushouts. The standard definition of excisive does lead to some very interesting mathematics, which you don't seem convinced by, so let me offer another perspective.
Consider a (homotopy) pushout square of spaces $$\require{AMScd} \begin{CD} A @>>> B \\ @VVV @VVV \\ C @>>> D \end{CD}$$ This says that the space $D$ can be built from some simpler pieces $A$, $B$, and $C$. The point of "excision" is that we would like to be able to be compute some (homotopy) invariants of $D$ from the invariants of these simpler pieces, but this fails for what is perhaps the most fundamental invariant of all: homotopy groups.
While homotopy groups don't interact well with pushouts, they do interact well with pullbacks. Namely, if $$\require{AMScd} \begin{CD} A @>>> B \\ @VVV @VVV \\ C @>>> D \end{CD}$$ were a pullback square, then we would get a long exact sequence relating their homotopy groups: $$\cdots \to \pi_k(A) \to \pi_k(B) \oplus \pi_k(C) \to \pi_k(D) \to \pi_{k-1}(A) \to \cdots$$
The failure of homotopy excision is one reason why we care about the identity functor on spaces not being excisive: pushout squares are generally not pullback squares, unlike in say an abelian or "stable" category.
On the other hand, there is an endofunctor $\operatorname{SP}^\infty$ that takes a space to its infinite symmetric power. It has the property that $\pi_k(\operatorname{SP}^\infty X) \cong H_k(X; \mathbb{Z})$ for a (connected) space $X$, and furthermore, this functor is excisive. So, given a pushout square as above, the square $$ \begin{CD} \operatorname{SP}^\infty A @>>> \operatorname{SP}^\infty B \\ @VVV @VVV \\ \operatorname{SP}^\infty C @>>> \operatorname{SP}^\infty D \end{CD}$$ is a pullback, so upon taking homotopy groups we obtain the usual long exact sequence in homology. Homology, or rather $\operatorname{SP}^\infty$, is excisive.
There are several other cool results related to these ideas. First, while pushout squares are not pullbacks in spaces, they are pullbacks "in a range" (depending on the connectivity of the spaces involved). This is the Blakers-Massey theorem. As the spaces become more and more connected, the range becomes larger and larger until we end up with "spectra", which in some sense is the excisive approximation of the category of spaces. Relatedly, we can try to write down the best approximation of the identity functor by an excisive functor, and this is the functor $Q = \Omega^\infty \Sigma^\infty$ you wrote down, which also passes through spectra.
Of course, this does not preclude studying functors that preserve homotopy pushouts, though those might not be very interesting - their values on any finite CW complex would be determined by their value on a point.