Scholar has the worst design of any job in the game, and here's why.

[Rilifane]

Lilies are not just heals. Lilies serve 3 purposes, they are used to move, they are used to weave because they are instant, and they are used to heal. WHM wants to use lilies to cover all three at once optimally. You can use lilies at max HP to move, because it gives you some damage return, and the potency return is higher than that of clipping your Dia. Ideally, you let your party stay lower on HP so that you can get some healing when you need to move. If your cohealer wants to be a heal bot that doesn't let you do that, then so be it, they can waste their resources for you playing correctly.

WHM's identity is literally its GCDs. WHM doesn't have the weaving space to properly use oGCD heals, and its entire design forces you to use GCDs. Its whole design is around using GCDs to heal, with things such as plenary rewarding you for doing them.

You missed the point. We all know that Lilies aren't just for healing like Medica II. But that's exactly the problem.
WHM has to use limited healing tools regardless of the situation to move OR weave and it's still a dps loss. SCH needs to use AF charges for dps neutral weaving/ moving but they have more per minute AND more healing tools that aren't tied to AF. AST LS was a MP tool before 5.3, it got changed though and now you're completely free to use delay it as much as you want for back to back mechanics, escpecially since you're not a card machine gun at the 3min marks anymore.

And healbotting isn't even the only reason why Lily healing during movement can be completely wasted. Boss design also plays a huge part. You can have a whole party plan everything around their WHM and they will still heal signifcantly less while overhealing more, meaning they are prone to wasting healing but not because they are a healing powerhouse that can throw out free heal like candy without any negative side effect. And that is with planning their movement and tools around one class.

And funny that you mention PI. It needs to be weaved before your GCD heals. And to weave it you either need a really opportune natural Dia refresh or - exactly, a Lily. That thing it's supposed to buff.

I just want to point out that even the most efficient, highest RDPS playres out there have an overheal % of around 20%.

Obviously.
Star and Assize are used on CD, Fairy keeps healing. Add some crit heals here and that last regen tick that wasn't even neccessary there and you have overheal even with a team of dedicated "How low can we go?" green dps. Plus sometimes there is the situation of one tool not being enough but two being too much.