In defence of Oliver, while a 115kg front squat is going to require a ton of glute strength, it's going to involve stuff all hamstring compared to a back squat, even a much lighter one than 115kg.
Now this is not so important for olympic lifters, when you're doing so many dynamic pulls and developing hamstring power that way. That, and the fact that the front squat is part of the clean & jerk, and actually part of the sport.
If your getting bad hip flexor pain from back squats then it's quite possible the cause is overactive/dominant quads or weak/inactive hamstrings. The two are reciprocal. Usually when someone says they have sore hip flexors from squatting it's not one of the true hip flexors (psoas), it's actually rectus femoris, one of the quads. In a low bar squat if the hamstring get loose at any stage, the rec fem pulls on the pelvis and produces tendonitis in the front of hip area. This exact problem prevented me from squatting for a long time. I'm still shit at squatting but I no longer get any hip flexor pain.
I suggest why front squats make everything feel better, is that the rectus femoris are getting a nice stretch at the bottom
Now, do you fix the underlying issue so that you can back squat pain free, which every healthy person should be able to do, or do you stick with what doesn't hurt, and potentially make the problem much worse by making your quads even more dominant?
Combining good mornings or RDLs or something for your hamstring won't necessarily do the trick either, because the quad doesn't do anything in those exercises. The hip flexor problem is when there is co-contraction of quad and hammy. Deep lunges, high weighted step-ups, ukrainian deadlifts off blocks or bulgarian split squats will train the hammies in that way, but I reckon if you can't back squat painfree, these exercises are likely to show up the same issues.
Short version, what NPR said.