Adverbial Accusative Phrases (1/2)

There are a number of fixed phrases in Latin that are accusative idiomatically. Here are a few:

Trēs hominēs id temporis exībam: I was dating three guys at that time.

Puellī molestiōrēs id aetātis fuerunt: Boys are a hassle at that age.

Quod genus anī est: What sort of old woman is she?

Meam vicem, habēas sicut placet: As far as I am concerned, you may have as much as you like.

Quid craterā hoc noctis agis: What are you doing with a mixing bowl at this hour of the night?

It isn’t clear to me how Allen and Greenough identify ‘quod (or id) genus‘ as an adverbial accusative. They agree it may have been a nominative at one point, but how do we know it isn’t a nominative now? Is it simply because ‘adverbial nominative’ isn’t an extant class?

The Essential AG: 397a