One exciting legacy of the Kepler mission was the discovery of about a dozen transiting circumbinary (CB) planets, despite potential barriers to planet formation in these systems. Determining the CB planet occurrence rate, and thus the efficiency of CB planet formation, depends on assumptions about the sensitivity of Kepler to CB planets as a function of the mutual inclination between the binary and planetary orbits, since it is a key parameter in determining whether and how frequently CB planets transit. To contextualize the known CB planet population, we examine young and intermediate age binary stars that host circumbinary protoplanetary and debris disks and investigate trends in system architecture—namely mutual inclination—with binary orbital parameters. We forward-model new ALMA observations of the pre-main sequence spectroscopic binary UZ Tau E and its CB disk using carbon monoxide J=2-1 emission. UZ Tau E joins V4046 Sgr, DQ Tau, and AK Sco as the fourth system whose stellar orbit and disk parameters are well-known and a direct calculation of the apparent mutual inclination between the stellar and orbital planes is possible. We find that all of these systems, which host short-period (P < 30 days) binary stars, have apparent mutual inclinations less than 3 degrees. The architectural similarity between these systems and those Kepler systems hosting CB planets suggests that the mutual inclinations of planets around short-period binaries are intrinsically low, implying that there does not exist a large population of misaligned CB planets orbiting binaries with these periods. We further extend our analysis to include all known circumbinary disk systems in the literature. This expanded sample includes many longer period binary stars, but suffers from incomplete stellar orbits and/or spatially unresolved disk observations, resulting in significant uncertainty in solutions of system architectures. Given the existing data, however, there are already several longer period (P ≿ 30 days) systems for which coplanarity is highly disfavored, such as KH 15D, GW Ori, and TWA 3A. This suggests that whatever formation mechanism or evolutionary pathway that produced the shorter period binaries tends to bring the binary orbital plane and the CB disk into alignment, and that its influence wanes with increasing binary orbital period.