= Solution
Start both systems from zero and compare their times to reach half of their \b[own] <Steady state>[steady concentrations]. Without regulation,
$$
Y(t)=\frac\beta\alpha(1-e^{-\alpha t}),\qquad
Y_{\rm st}^{(0)}=\frac\beta\alpha,\qquad T_{1/2}^{(0)}=\frac{\log2}{\alpha}.
$$
In the strong <logic approximation for gene regulation>, <Gene expression>[production] is initially fully on and the <concentration> rises towards the repression threshold. The regulated level is pinned near $Y_{\rm st}\simeq K$, so it reaches $K/2$ before switching:
$$
\boxed{T_{1/2}^{\rm logic}=-\frac1\alpha\log\left(1-\frac{\alpha K}{2\beta}\right),\qquad
\frac{T_{1/2}^{\rm logic}}{T_{1/2}^{(0)}}=
\frac{-\log[1-\alpha K/(2\beta)]}{\log2}
\simeq\frac{\alpha K}{2\beta\log2}\ll1.}
$$
The faster relative response comes from allowing high initial <Gene expression>[production] while keeping the final abundance low. <Negative autoregulation> can reduce response delays and <Sensitivity analysis>[sensitivity] to <Gene expression>[production] fluctuations, and can reduce unnecessary accumulation of a <protein>. It does not suppress every possible noise source at every frequency.
A perfectly discontinuous on/off right-hand side has a threshold-switching ambiguity rather than an ordinary smooth equilibrium at $K$. Here threshold pinning means the limiting rapid-switching or steep smooth-response idealization. A continuous repression law resolves that idealization. Comparing half of the same prescribed <concentration>, rather than half of each model's final <concentration>, would be a different response-time comparison.
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