It's wrong because much of the data is not consistent with your idea if you do the math. The data presented in Leathers et al 1998 is useful here as they consider how Tmax varies with DTR. Even for those with a linear Tmax to DTR relationship, these don't have consistent percentage drops (and so your explanation doesn't work). Additionally, as described in the original answer, many sites have very complicated relationships between Tmax and DTR and with variable timing of seasonal maxes of DTR. For example, one site where at days with a Tmax of nearly 30 C and those with a Tmax of -5 C have the same DTR of ~12 C (with a minimum in DTR with a Tmax of 5 C). I.e., looking at actual data highlights that your explanation is not useful as a broad explanation.