The SFF's linear ramp regime is a fingerprint of chaos because it results from level repulsion. The eigenvalues of the Hamiltonian (or unitary evolution operator) in a chaotic (quantum) system repel one another due to interactions.
This is also captured by the $r$ ratio for level statistics, which is the average ratio between consecutive energy differences $r = \min(\Delta_i,\Delta_{i+1})/\max(\Delta_i,\Delta_{i+1})$, where $\Delta_{i} = E_i - E_{i-1}$. We find $r \sim 0.38$ for Poisson-distributed eigenvalues and $r \sim 0.55$ or something for Gaussian-distributed eigenvalues.
As you say, integrable systems have Poisson statistics, since they have an extensive number of conserved quantities; as a result, the eigenvalues basically all belong to independent "charge" sectors, and thus do not repel one another. So, we should not expect to see $K(t) \sim t$ either!
So what does the SFF look like for a nonchaotic system? Well, there shouldn't be a linear ramp regime!
For small systems integrable systems, you might see only a dip and plateau, and no linear regime. This happens if you plot the SFF for a single qubit, which looks like $K(t) = 2 + e^{-t^2} (2 - 4t^2)$ when $H$ is drawn from the Gaussian unitary ensemble (after averaging). This doesn't really have a linear regime.
For perfectly uncorrelated systems, you will see $K(t) = \mathcal{D}$ for $t > 0$ where $\mathcal{D}$ is the Hilbert space dimension (note that $K(0) = \mathcal{D}^2$, by definition). On the other hand, for a chaotic system, we only have $K(t) = \mathcal{D}$ for $t > \mathcal{D}$, for reasons I explain in this answer on SFFs. In fact, if you consider a model of noninteracting qubits hopping on a lattice with random $Z$-field disorder, averaging the SFF over disorder gives exactly $K(t)= \mathcal{D}$ for $t >0$. This model is not integrable, of course, but an Anderson insulator.
More generally, as you note in your question and I also note in my previous answer, systems with conservation laws (or systems that are not maximally chaotic) only realize the linear ramp regime after some time $t \geq t^{\,}_{\rm th}$, which is the "Thouless time" (see this paper by John Chalker). This is the time at which chaos sets in, and it follows the "dip" regime you identified. In general, a system fails to be chaotic if $t^{\,}_{\rm th} > \mathcal{D}$.
So basically, any integrable system either (1) clearly doesn't have a linear ramp regime or (2) analytical calculations will predict the ramp regime starts after $t=\mathcal{D}$, so it does not actually exist. In practice, the prediction for integrable systems is roughly $K(0)=\mathcal{D}^2$ and $K(t>0) \sim \mathcal{D}$, with no significant functional dependence on $t$ (though some fluctuations and dip-like behavior may manifest).
