Does a technical umbrella term exist that covers the different kinds of biomolecules?
I don’t know what your background is. Maybe it is organic chemistry, where the term functional group is used in a very specific way and using what would seem to be a synonym instead is wrong. Maybe it is biology (taxonomy) where it makes a big difference if a ‘group’ is an order, a family, a genus or something else.

The typical primary metabolites in biochemistry — carbohydrates, peptides/proteins, nucleic acids and lipids — do not have an overarching, widely accepted technical umbrella term. You can call them groups, classes, kinds or even types and be well understood.

If I had to suggest something, I would go with classes of primary metabolites. But I wouldn’t call that a strict technical term like functional group (organic chemistry) or species (taxonomy).