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It is shown that a large class of systems of non-linear wave equations, based on the good-bad-ugly model, admit formal solutions with polyhomogeneous expansions near null infinity. A particular set of variables is introduced which allows us to write the Einstein field equations in generalized harmonic gauge as a good-bad-ugly system and the functional form of the first few orders in such an expansion is found by applying the aforementioned result. Exploiting these formal expansions of the metric components, the peeling property of the Weyl tensor is revisited. The question addressed is whether or not the use of generalized harmonic gauge, by itself, causes a violation of peeling. Working in harmonic gauge, it is found that log-terms that prevent the Weyl tensor from peeling do appear. The impact of gauge source functions and constraint additions on the peeling property is then considered. Finally, the special interplay between gauge and constraint addition, as well as its influence on the asymptotic system and the decay of each of the metric components, is exploited to find a particular gauge which suppresses this specific type of log-term to arbitrarily high order.