Abundance estimates from animal point-count surveys require accurate estimates of detection probabilities. The standard model for estimating detection from removal-sampled point-count surveys assumes that organisms at a survey site are detected at a constant rate; however, this assumption can often lead to biased estimates. We consider a class of N-mixture models that allows for detection heterogeneity over time through a flexibly defined time-to-detection distribution (TTDD) and allows for fixed and random effects for both abundance and detection. Our model is thus a combination of survival time-to-event analysis with unknown-N, unknown-p abundance estimation. We specifically explore two-parameter families of TTDDs, e.g., gamma, that can additionally include a mixture component to model increased probability of detection in the initial observation period. Based on simulation analyses, we find that modeling a TTDD by using a two-parameter family is necessary when data have a chance of arising from a distribution of this nature. In addition, models with a mixture component can outperform non-mixture models even when the truth is non-mixture. Finally, we analyze an Ovenbird data set from the Chippewa National Forest using mixed effect models for both abundance and detection. We demonstrate that the effects of explanatory variables on abundance and detection are consistent across mixture TTDDs but that flexible TTDDs result in lower estimated probabilities of detection and therefore higher estimates of abundance.