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Current cosmological data exhibit a tension between inferences of the Hubble constant, $H_0$, derived from early and late-universe measurements. One proposed solution is to introduce a new component in the early universe, which initially acts as "early dark energy" (EDE), thus decreasing the physical size of the sound horizon imprinted in the cosmic microwave background (CMB) and increasing the inferred $H_0$. Previous EDE analyses have shown this model can relax the $H_0$ tension, but the CMB-preferred value of the density fluctuation amplitude, $σ_8$, increases in EDE as compared to $Λ$CDM, increasing tension with large-scale structure (LSS) data. We show that the EDE model fit to CMB and SH0ES data yields scale-dependent changes in the matter power spectrum compared to $Λ$CDM, including $10\%$ more power at $k = 1~h$/Mpc. Motivated by this observation, we reanalyze the EDE scenario, considering LSS data in detail. We also update previous analyses by including $Planck$ 2018 CMB likelihoods, and perform the first search for EDE in $Planck$ data alone, which yields no evidence for EDE. We consider several data set combinations involving the primary CMB, CMB lensing, SNIa, BAO, RSD, weak lensing, galaxy clustering, and local distance-ladder data (SH0ES). While the EDE component is weakly detected (3$σ$) when including the SH0ES data and excluding most LSS data, this drops below 2$σ$ when further LSS data are included. Further, this result is in tension with strong constraints imposed on EDE by CMB and LSS data without SH0ES, which show no evidence for this model. We also show that physical priors on the fundamental scalar field parameters further weaken evidence for EDE. We conclude that the EDE scenario is, at best, no more likely to be concordant with all current cosmological data sets than $Λ$CDM, and appears unlikely to resolve the $H_0$ tension.