This article presents a measurement of $\nu_e$ interactions without pions in the final state using the MicroBooNE experiment and an investigation into the excess of low-energy electromagnetic events observed by the MiniBooNE collaboration. The measurement is performed in exclusive channels with (1$e$N$p$0$\pi$) and without (1$e$0$p$0$\pi$) visible final-state protons using 6.86$\times 10^{20}$ protons on target of data collected from the Booster Neutrino Beam at Fermilab. Events are reconstructed with the Pandora pattern recognition toolkit and selected using additional topological information from the MicroBooNE liquid argon time projection chamber. Using a goodness-of-fit test the data are found to be consistent with the predicted number of events with nominal flux and interaction models with a $p$-value of 0.098 in the two channels combined. A model based on the low-energy excess observed in MiniBooNE is introduced to quantify the strength of a possible $\nu_e$ excess. The analysis suggests that if an excess is present, it is not consistent with a simple scaling of the $\nu_e$ contribution to the flux. Combined, the 1$e$N$p$0$\pi$ and 1$e$0$p$0$\pi$ channels do not give a conclusive indication about the tested model, but separately they both disfavor the low-energy excess model at $>$90% CL. The observation in the most sensitive 1$e$N$p$0$\pi$ channel is below the prediction and consistent with no excess. In the less sensitive 1$e$0$p$0$\pi$ channel the observation at low energy is above the prediction, while overall there is agreement over the full energy spectrum.