The severe shortfall in testing supplies during the initial phases of the COVID-19 pandemic and ensuing struggle to control disease spread have affirmed the need to plan rigorous optimized supply-constrained resource allocation strategies for the next inevitable novel disease epidemic. To address the challenge of optimizing limited resource usage in the face of complicated disease dynamics, we develop an integro partial differential equation disease model which incorporates realistic latent, incubation, and infectious period distributions along with limited testing supplies for identifying and quarantining infected individuals, and we analyze the influence of these elements on controllability and optimal resource allocation between two testing strategies, `clinical' targeting symptomatic individuals and `non-clinical' targeting non-symptomatic individuals, for reducing total infection sizes. We apply our model to not only the original, delta, and omicron COVID-19 variants, but also to generic diseases which have different offsets between latent and incubation period distributions which allow for or prevent varying degrees of presymptomatic transmission or preinfectiousness symptom onset. We find that factors which reduce controllability generally call for reduced levels of non-clinical testing, while the relationship between symptom onset, controllability, and optimal strategies is complicated. Although greater degrees of presymptomatic transmission reduce disease controllability, they may enhance or reduce the role of non-clinical testing in optimal strategies depending on other disease factors like overall transmissibility and latent period length. Our model allows a spectrum of diseases to be compared under the same lens such that the lessons learned from COVID-19 can be adapted to resource constraints in the next emerging epidemic and analyzed for optimal strategies under a consistent mathematical framework.