> you just can't use By the time this way

I worked like hell to get a doctorate in English decades ago, and this usage seems absolutely normal to me.

In "theory" (which I've only just now discovered online - or, I should say, "on line"), "by the time" should only be used with a past perfect verb.

But the sad, or totally innocuous, fact (depending on one's prejudices) is that the simple past often - very, very often - replaces the past perfect in other than very formal usage, especially when context makes the distinction immaterial. The meaning in the given example couldn't be clearer.