With birds in which the chicks are helpless and confined to the nest for weeks after hatching (altricial), they need sheltering, defending from predators, frequent feeding and their droppings taken away. It's a two-bird job, so if a male shags and immediately leaves to court another female, there's pretty much zero chance that any chicks he fathers will survive. However, sneaky extra-pair copulations between birds of neighbouring territories are VERY common among some of these supposedly 'monogamous' species.
You see obvious sexual dimorphism more often in species with precocial chicks, able to run about and feed themselves soon after hatching (eg ducks, chickens). The female (or in a few cases the male - eg in phalaropes and Dotterels the male does all the parental care) doesn't need a second helper so can select a mate solely on the basis of looks and vigour.
Edited by aiki, : clarify some bits