Remember, always take Carrier with buckets of salt, as he is wont to misrepresent and ignore evidence inconvenient to him.
In this case, the vast majority of Academia see Luke/Acts and Josephus as separate sources. If I recall, notables like Theodor Mommsen and Thomas Goud wrote on Josephus, and used Acts to cross-reference, for instance. It is true that either they are separate and referencing the same source or events, or one is dependant on the other; but the argument of Acts using Josephus is a minority view that has deep problems.
Firstly, they have a lot in common, but mostly they record different events. What they agree on are expulsions of Jews under Claudius, some governors, a few noted rebellions, etc. That is hardly sufficient to establish a relationship, as those were just current events - as if two texts mentioning Winston Churchill and some of WWII had to crib from one another. Just having a figure mentioned such as Judas the Galilean, hardly equates to firm evidence of derivation.
But the kicker for this to work, is you have to assume Josephus corrupted - as their timelines differ a bit (for instance, assume the Isis scandal an insertion).
There are also big differences - most notably, Pilate's title. Josephus with Tacitus use Procurator. Luke uses Prefect. Historically, this had been used to denigrate Luke as inacurate, until we discovered archaeologically that Pilate's title was Prefect - and those two historians were wrong. For in 42 AD when Judaea was reconstituted by Claudius after the end of Herod Agrippa's client state, the governor was thereafter a Procurator; but prior to this, in the province from 6 to 38 AD, it had been Prefect it seems. This is evidence that Luke has legitimate information deriving from prior to 38 AD, regardless when actually written. It also shows that Josephus as a source is unlikely, or it is more than likely that Luke would have used the wrong title for Pilate.
The only reason Luke is placed after 70 AD is due to the assumption that if a prophecy mentions something, it had to have been written after it. This is nothing but presupposition, as Simpsons has a President Trump years before he took office. Even if you don't believe in Prophecy, Judaea was a restive province, so a prediction that Rome would someday destroy Jerusalem as they destroyed Corinth or Carthage or Numantia, is really not far-fetched. In my opinion, this dating is simply bias, in that people do not want early dates for Gospels, as it messes with the whole 'legend' explanation beloved by secular scholarship. In the 19th century they argued for 2nd or 3rd century dates, till forced to place it earlier and earlier. This is just the forlorn hope of a hypothesis, that people are grasping at straws to maintain.
