Effective Capacitive Loading of Active RC Integrator
But it doesn't really make sense since \$C_I\$ is charged by input current, not the opamp, so the opamp doesn't "see" the integrating capacitor.

Incorrect, the input current flows into the resistor, through the node Vx, through the capacitor and into the op-amp output. Therefore the op-amp load is \$C_I\$.