Not an answer, but maybe some intuition from a tangential direction, and too long for a comment. I hope the information here can be useful nonetheless.
We can consider a somewhat more geometric situation, algebraic K-theory and Witt groups for smooth schemes over fields of characteristic $\neq 2$ to get some intuition what could happen. Also, trivial involution. In this case, algebraic K-theory and Witt groups are representable by spectra $\mathbf{KGL}$ and $\mathbf{KT}$,respectively.
On the one hand, I would like to think of the following things as indication that Witt groups could be nontrivial even if K-groups vanish. One possibility to link algebraic K-theory and Witt groups is via hermitian K-theory, aka higher Grothendieck-Witt theory. There are Karoubi periodicity sequences linking algebraic and hermitian K-theory by means of forgetful and hyperbolic functors. In terms of the representing spectra, these can be formulated as cofiber sequences
$$
\Sigma^{1,1}\mathbf{KO}\xrightarrow{\eta}\mathbf{KO}\to\mathbf{KGL}
$$
This is explained in
- O. Röndigs and P.A. Østvær. Slices of hermitian K-theory and Milnor's conjecture on quadratic forms. Geom. Topol. 20 (2016), no.2, 1157--1212. doi-link to paper
From this viewpoint, the vanishing of algebraic K-theory implies that $\eta$ is invertible on hermitian K-theory. But the Karoubi tower construction of $\mathbf{KT}$ means $\mathbf{KT}=\mathbf{KO}[\eta^{-1}]$, so the Witt-theory spectrum is actually obtained by making $\eta$ invertible on hermitian K-theory. I would interpret this as saying that the spectrum ${\bf KT}$ is less linked to algebraic K-theory than the rest of hermitian K-theory.
In a similar vein, one can consider the real realization of the relevant spectra, for example discussed in
The real realization of $\mathbf{KGL}$ is trivial, while the real realization of hermitian K-theory and Witt theory are both $\mathbf{KO}^{\rm top}[1/2]$. In some sense, real realization doesn't care about algebraic K-theory, and the hermitian K-theory information that is visible in the real realization is coming from Witt theory. This seems to say that it's possible to have situations where the Witt groups of $S$ could be nontrivial even when algebraic K-theory is trivial.
On the other hand, one can consider the slice spectral sequence for $\mathbf{KT}$. A discussion of the spectral sequence can also be found in the Röndigs-Østvær paper linked above. The spectral sequence has the form
$$
E^1_{p,q,n}=\pi_{p,n}\left(\bigvee_{i\in\mathbb{Z}}\Sigma^{2i+q,q}\mathbf{MZ}/2\right)\Rightarrow\pi_{p,n}\mathbf{KT}
$$
This is a method to compute the Witt theory from mod 2 motivic cohomology (with the motivic Steenrod operations appearing as first differentials). A consequence of the spectral sequence is that vanishing of mod 2 motivic cohomology implies vanishing of Witt groups.
It's not quite clear to me what to expect in the somewhat different situation of algebraic K-theory and Witt groups for stable $\infty$-category. The first part, linking Witt groups and algebraic K-theory, probably works in somewhat greater generality, while the second part seems to be quite tied to a geometric situation.
