I'm a level 28 Wood elf. I lost Erik the Slayer to Alduin.

I decided I wanted to get J'zargo. So I went to the College. I met him and we pvssyd. He didn't give me any quest option to do for him. I thought perhaps I needed to do a bit for the College first, so he could see me in action. So I did the first quest for the College. He still doesn't give me an option.

Is this a bug, or am I missing something?

just keep on with the CoW questline, and checking with him periodically - he will eventually offer you his quest.

So this is by design? Whew.

I feared I had another bug on my hands.

No, it took a little while for him to offer me the quest too. I cant remember exactly where I was up to, but it was after the first few CoW quests, maybe check with him after each stage.

Make sure you keep telling him you're going to make him into a powerful mage.

You can start J'zargo's quest when you get the quest to return the books to the college ("Hitting the Books").