I am running tanks and systems without skimmers for decades. At the moment we run 2 systems of two tanks each and two single tanks without skimmer and only one system with skimmer, a DIY floating skimmer from a modified old Tunze horizontal model which works great. This system always has the highest phosphate concentrations, sometimes almost twice as high as the next highest phosphate concentration in a system or tank.
Scientific findings of bubbling ocean water or saltwater say exopolymers/slime of algae (and in my opinion bacteria) adsorb to air bubbles and have
C : N ratios of ca. the Redfield ratio. Nitrogen and organic carbon limitation of bacteria may be the cause of increased phosphate concentrations in our skimmed system.
By the way, proteins need nitrogen.