In terms of appropriateness, do (since + past) and (since + perfect) have different meaning and usage?

Question

This problem is discussed in Advanced Oxford Grammar and the book explains that different tenses are possible in sentences with since. By the way, I quoted the explanations and some examples below:

There may be a past tense in the time expression after since.

E.g. We've lived here since we got married [1].

A present perfect is also possible in the time expression, to talk about continuation up to now.

E.g. We've lived here since we've been married [2].

As for my understanding after reading the information above, my first thinking was they are interchangeably used. However, the exercise in the book says otherwise. To illustrate this, I also provide the example taken from the exercise.

We've all been eating much better since the new chef __ doing the cooking.

1. has been (v)
2. was (x)
3. both (x)

The key answer tells that the answer is has been. This is where I'm starting to be confused. I mean, the explanations of the book I quoted above, in my opinion, imply that both perfect and past are possible. Then, how come the answer is not both?, meaning this sentence should be true?:

We've all been eating much better since the new chef was doing the cooking.

P.S. The book doesn't tell the different usage.

Answer 1

The answer should be has been, because the time frame of the sentence is set by the verb have ... been eating.

The meaning with have been eating and has been doing the cooking is that both time periods coincide -we started eating well when he started cooking, and he still cooks and we still eat well.

The other answer since the new chef did the cooking isn't ungrammatical, but the meaning is very unlikely.

But you can change the meaning so that past tense makes sense:

We've all been eating much better since the new chef changed the menu.