We characterize the pH-controlled polymer capture and transport through silicon nitride (SiN) pores subject to protonation. A charge regulation model able to reproduce the experimental zeta potential of SiN pores is coupled with electrohydrodynamic polymer transport equations. The formalism can quantitatively explain the experimentally observed non-monotonic pH dependence of avidin conductivity in terms of the interplay between the electroosmotic and electrophoretic drag forces on the protein. We also scrutinize the DNA conductivity of SiN pores. We show that in the low-pH regime where the amphoteric pore is cationic, DNA-pore attraction acts as an electrostatic trap. This provides a favorable condition for fast polymer capture and extended translocation required for accurate polymer sequencing.