your drawing is wrong, the photodiode current goes into the op amp -input so that it is the servoed variable, do look in the Vishay app note 50 fig 18
the LED drive is just using the PNP as a unity gain current buffer for the op amp output V in the correct version of the circuit
since you are supposed to measure, servo the actual current from the photodiode the entire LED drive circuit is inside the feedback loop, there is next to zero consequence of the PNP buffer Vbe or hfe variations if the op amp gain open loop gain is high
if the op amp can provide the required current directly, allowing for the LED and current limit R V drops, the supply V and the op amp output Vsat, then you can lose the Q - see the other circuit examples in the app note