  Massive, rapidly-spinning magnetar remnants produced as a result of binary
neutron star (BNS) mergers may deposit a fraction of their energy into the
surrounding kilonova ejecta, powering a synchrotron radio signal from the
interaction of the ejecta with the circumburst medium. We present 6.0 GHz Very
Large Array (VLA) observations of nine, low-redshift short gamma-ray bursts
(SGRBs; $z<0.5$) on rest-frame timescales of $\approx2.4-13.9$ yr following the
bursts. We place $3\sigma$ limits on radio continuum emission of
$F_{\nu}\lesssim6-20\,\mu$Jy at the burst positions, or
$L_{\nu}\lesssim(0.6-8.3)\times10^{28}$erg s$^{-1}$Hz$^{-1}$. Comparing these
limits with new light curve modeling which properly incorporates relativistic
effects, we obtain limits on the energy deposited into the ejecta of
$E_{ej}\lesssim(0.6-6.7)\times 10^{52}$erg
($E_{ej}\lesssim(1.8-17.6)\times10^{52}$erg) for an ejecta mass of
$0.03\,M_{\odot}$ ($0.1\,M_{\odot}$). We present a uniform re-analysis of 27
SGRBs with $5.5-6.0$ GHz observations, and find that $\gtrsim50\%$ of SGRBs did
not form stable magnetar remnants in their mergers. Assuming SGRBs are produced
by BNS mergers drawn from the Galactic BNS population plus an additional
component of high-mass GW194025-like mergers in a fraction $f_{GW190425}$ of
cases, we place constraints on the maximum mass of a non-rotating neutron star
(NS) ($M_{TOV}$), finding $M_{TOV}\lesssim2.23\,M_{\odot}$ for
$f_{GW190425}=0.4$; this limit increases for larger values of $f_{GW190425}$.
The detection (or lack thereof) of radio remnants in untargeted surveys such as
the VLA Sky Survey (VLASS) could provide more stringent constraints on the
fraction of mergers that produce stable remnants. If $\gtrsim30-300$ radio
remnants are discovered in VLASS, this suggests that SGRBs are a biased
population of BNS mergers in terms of the stability of the remnants they
produce.
