It was indeed the case at the time. The Job Stones worked by giving you new spells after level 30, exactly one new skill every 5 levels, so SMN and SCH shared the exact same skillset until lvl 30 and even then, they shared ACN skills up until 50, this meant that SCH shared Ruin 1 and 2 and both levels of Bio and Miasma, which incidentally made SCH the more "DPS-oriented" healer, SMN still has Physick I don't know if you mean SCH gets a Scholar-only version or something because I haven't seen SCH's skillset in a while (it was certainly useless for SMN since HW, because it scales off MND and not INT), I'd guess they made SCH Ruin also scale off MND instead of INT for SMN with the removal of Cleric Stance.

Yes, this indeed created a huge problem when one of your healer jobs couldn't use any of its actual healing skills bar Physick pre-Haukke Manor, and meant you had to be spamming Physick in Brayflox due to the boss's undodgeable poison DoT. However, the fairy is otherwise so OP in that content that a SCH didn't really need to HEAL because the fairy would take care of everything. (another fun tidbit was that Cleric Stance swapped your MND and INT attributes, and your fairy's heals were based on your MND at the time of summoning, so you could summon her without CS, then turn it on and DPS, and she wouldn't be affected; however, summon her with CS ON and you'd nerf her heals for no reason)

So, yes, SCH was kind of a mess in ARR up to ShB because of the Job System and being the only job that shares the same base class, while also being the only job that's a whole different ROLE than its base class.