As explained at length in prior threads, a test statistic is any statistic that you test. That's all it means. There's not one per model or something. Certain test statistics may have additional nice properties (like generating correct p-values) but this isn't necessary for the definition.

Casella & Berger give an example when they first introduce the idea. They say take a mean. Now compare it to the number 5. That's using the mean as a test statistic. Won't have good properties, but meets the definition. So once you understand that, it's trivially obvious that a coefficient can be a test statistic, just as any statistic can be. C&B even have a problem where they ask you to show that a coefficient, when used as a test statistic, will generate the same p-values as using the t statistic.

-- A public service announcement from someone who actually understands stats.