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 $3sigma$ limits on radio continuum emission of $F_{ u}lesssim6-20,mu$Jy at the burst positions, or $L_{ u}lesssim(0.6-8.3)times10^{28}$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.
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