We establish a unified analytical framework for load balancing systems, which allows us to construct a general class $Π$ of policies that are both throughput optimal and heavy-traffic delay optimal. This general class $Π$ includes as special cases popular policies such as join-shortest-queue and power-of-$d$, but not the join-idle-queue (JIQ) policy. In fact, we show that JIQ, which is not in $Π$, is actually not heavy-traffic delay optimal. Owing to the significant flexibility offered by class $Π$, we are able to design a new policy called join-below-threshold (JBT-d), which maintains the simplicity of pull-based policies such as JIQ, but updates its threshold dynamically. We prove that JBT-$d$ belongs to the class $Π$ when the threshold is picked appropriately and thus it is heavy-traffic delay optimal. Extensive simulations show that the new policy not only has a low complexity in message rates, but also achieves excellent delay performance, comparable to the optimal join-shortest-queue in various system settings.