  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$\sigma$
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.
