It seems to me that, despite several systems developed for higher-order logics, almost all the attention in Logic is devoted to first-order. I understand that higher-order logics have some drawbacks, such as the compactness theorem and the Löwenheim-Skolem theorems and other such not holding in second-order. However, there are some advantages as well, for example doing away with a lot of messy schemata. I would understand if the whole Logic community was composed of Intuitionists or Constructivists, since sets/classes using quantification over sets or relations are harder to construct, or if the Logic community was worried about programming issues, but since neither of these is in fact the case, I am puzzled by the preponderance of attention to first-order results. I know this is a broad question, so any responses will be appreciated.