Auberney's suggestion is similar to the Race/Class system in D+D or DW, except that it's 50/50. Possible, but it would be easier (not necessarily better) for, like D&D, the race to provide some kind of stat boost, extra move, other modifier, or some combination of those three, and the player is still primarily their class/role. Perhaps all Gorillas have +1 Strength and access to the move "Smash Things", while Orangutans have +1 Intelligence and "Puzzle it Out".
Another option would be to do the regular class-only playbooks, but make sure each race is represented in more than one playbook. You could have let's say the Berserker and Guardian for gorillas, the Handychimp and Sneaker for chimps, and the Elder and Doctor for orangutans.
If a player wants to make a character who was a hybrid of more than one idea, that's what multiclassing (IE "take a move from another playbook") is for. For "ape moves", couldn't you just add them to the list of basic moves? Characters are already somewhat restricted in what basic moves they can make useful just based on their stats, so it might not be necessary to make players put skill points into them to restrict their use.
What was the problem you ran into with role-based playbooks?