To facilitate the use of population counts as an index of population change, we describe a generalization of the distance sampling methodology to analyze, in addition to distance to the observer, two other ways to estimate imperfect detection probability: multiple observers and time-to-detection, in a flexible manner, meaning that not all sites or years need to have distance information or be surveyed in the same way every year. We also account for the effect of partially-observed individual covariates, to account for the effect of group size on detection probability. Finally, we separate the probability of availability to detection from the probability of detection itself. We perform a thorough, illustrated assessment of the pros and cons of this framework with simulations and real case studies. First, we compare to simple linear models, illustrating the magnitude of the bias caused by imperfect detection. Second, we compare to standard distance sampling, illustrating the bias caused by variation in the probability of availability to detection. However, the availability to detection was weakly identifiable, meaning that the ability to separate it from detection probability, and therefore debias the trend estimate, depended on the data configuration. Combining distance with multiple observers and with time-to-detection solved the weak identifiability in an applied case study. We recommend using both the type of analysis we showcase, and a simple regression of the population count against time. Discrepancies between results from simple and complex analyses can help identify sources of bias in the former and loss of precision in the latter within the logistical constraints of local wildlife management schemes.