Add proficiency bonus to AC.
Double the effect of healing on self when in beast form, and allow casting abdurations/healing as a bonus action. (This might fix the spell slot issue mentioned by @Chaosmancer)
Gain inspiration when wildshaping - optional rule this can be replaced by DM granting proficiency/experise in an form aproperiate skill.
Make the moon druid bonus action a bestial strike rather than an unarmed strike (i suspect this was what was intended)
I think these are ideas that would at least start working toward a solution to the main criticisms (except tiny) while keeping the core ideas?
If that is your intend, and your proposal of subclass at level 1 is in place, that could work.
But since I like it better that you play your class for 2 levels before making that big decision, I think, putting a fixed STR in (and making that higher for moon druids) makes more sense.
I think, if you would make a 5e derrivate game, that has subclass 1st level which impact the base classes way more than they do now, it would make a fine game... but I think OneD&D is the wrong place.