The parabolic–elliptic version of the logistic Keller–Segel system given by $$\begin{aligned} \left\{ \begin{array}{lcll} u_t &{}=&{} \Delta u - \chi \nabla \cdot (u\nabla v) + \kappa u - \mu u^2, &{}\quad x\in \varOmega , \quad t>0, 0 &{}=&{} \Delta v - m(t) + u, \qquad m(t):=\frac{1}{|\varOmega |} \int _\varOmega u(x,t) \mathrm{d}x, &{}\quad x\in \varOmega , \quad t>0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$is considered in the ball $$\varOmega =B_R(0)\subset {\mathbb {R}}^n$$ with $$n\ge 1$$ and $$R>0$$, and with parameters $$\kappa \in {\mathbb {R}}$$, $$\chi >0$$ and $$\mu >0$$. The focus is on the question how the zero-order dissipative term $$-\,\mu u^2$$ herein, forming the apparently most essential difference between ($$\star $$) and the classical parabolic–elliptic Keller–Segel system, affects the evolution of supposedly present singular structures. For this purpose, a Neumann-type initial boundary value problem for ($$\star $$) with $$\mu >\chi $$ is studied for radially decreasing nonnegative initial data $$u_0\in C^1(\overline{\varOmega }{\setminus }\{0\})$$ fulfilling $$u_0(x) \le K\phi (|x|)$$ for all $$x\in \overline{\varOmega }{\setminus } \{0\}$$ with some $$K>0$$ and some function $$\phi : (0,\infty )\rightarrow [1,\infty )$$ which, besides some technical assumptions, complies with the key condition $$\begin{aligned} \int _0^1 r^{n-1} \ln \phi (r) \mathrm{d}r <\infty . \end{aligned}$$It is seen that for this class of data, including any such $$u_0$$ satisfying $$\begin{aligned} u_0(x) \le K e^{\lambda |x|^{-\alpha }} \qquad \text{ for } \text{ all } x\in \overline{\varOmega }{\setminus } \{0\} \end{aligned}$$with some positive constants $$K, \lambda $$ and $$\alpha <n$$, the problem in question in fact admits a global solution (u, v) which is smooth and classical in $$\overline{\varOmega }\times (0,\infty )$$ and attains the initial data in the topology of $$C^0_{\mathrm{loc}}(\overline{\varOmega }{\setminus } \{0\})$$ as $$t\searrow 0$$. In view of the well-known fact that in the unperturbed Keller–Segel system already some finite-mass Radon measure-type singularities give rise to persistently singular solutions, and that hence no significant smoothing action can be expected there when infinite-mass distributions are initially present, these results reveal that zero-order quadratic degradation indeed may have a substantial effect on cross-diffusive interaction by enforcing instantaneous smoothing even of initial data exhibiting some exponentially strong singularities.