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The dividing line between gamma-ray bursts (GRBs) and ordinary stripped-envelope core-collapse supernovae (SNe) is yet to be fully understood. Observationally mapping the variety of ejecta outcomes (ultra-relativistic, mildly-relativistic or non-relativistic) in SNe of Type Ic with broad lines (Ic-BL) can provide a key test to stellar explosion models. However, this requires large samples of the rare Ic-BL events with follow-up observations in the radio, where fast ejecta can be probed largely free of geometry and viewing angle effects. Here, we present the results of a radio (and X-ray) follow-up campaign of 16 SNe Ic-BL detected by the Zwicky Transient Facility (ZTF). Our radio campaign resulted in 4 counterpart detections and 12 deep upper limits. None of the events in our sample is as relativistic as SN 1998bw and we constrain the fraction of SN 1998bw-like explosions to $< 19\%$ (3$σ$ Gaussian equivalent), a factor of $\approx 2$ smaller than previously established. We exclude relativistic ejecta with radio luminosity densities in between $\approx 5\times10^{27}$ erg s$^{-1}$ Hz$^{-1}$ and $\approx 10^{29}$ erg s$^{-1}$ Hz$^{-1}$ at $t\gtrsim 20$ d since explosion for $\approx 60\%$ of the events in our sample. This shows that SNe Ic-BL similar to the GRB-associated SN 1998bw, SN 2003lw, SN 2010dh, or to the relativistic SN 2009bb and iPTF17cw, are rare. Our results also exclude an association of the SNe Ic-BL in our sample with largely off-axis GRBs with energies $E\gtrsim 10^{50}$ erg. The parameter space of SN2006aj-like events (faint and fast-peaking radio emission) is, on the other hand, left largely unconstrained and systematically exploring it represents a promising line of future research.