fields are not algebraic
Fields do not share nice properties with algebraic categories such as rings and groups because it isn’t one. In fact, rings and groups are monadic (a slightly stronger notion than algebraic, according to the definitions of some authors) but field are not. This is easily seen because fields do not have initial or terminal objects.
This is maybe a good perspective to see why there is a relatively richer study of field extension compared to general fields. The category of field extensions over some field is algebraic (?).