I don't understand this question. I don't see how we could possibly test the hypothesis that T(t) = a + U(t), without estimating b. What exactly do we compute and how do we decide that T(t) is "just a" or not "just a"?

The only way to deal with the question of trend is, in my view, is to assume that T(t) = a + b* t + U(t) and to estimate "b". We can also assume other models, e.g. T(t) = a + b*t + c*t*t + U(t), etc.