I've been looking at "An Aramaic approach to Q", and although I've not read the whole thing, and could cover parts in more detail, I have at least an initial assessment.
I think he put his best stuff first. In Luke 11:39-51 there are signs of Aramaic, and the sitz in leben does seem plausibly historical. That Luke's "rue" and Matthew's "dill" differ by one pen stroke in Aramaic stands out. After that things go downhill, in my opinion. I found the treatment of Mark 3-20:30 particularly problematic. He proposes a missing Greek Q and a missing Aramaic Q, and a missing Aramaic Mark, (It seems everywhere he proposes an Aramaic connection, he also proposes a Greek connection between Matthew and Luke ) and gives us little if anything that would argue against the idea that Matthew used Mark and only Mark here, and Luke used Matthew and only Matthew here. The historical sitz in leben he proposes has the historical Jesus performing dramatic exorcisms. (Well maybe but
To me, while it does seem there is some Aramaic floating around in the air perhaps, I don't think we have enough evidence for an Aramaic source behind the double tradition, at least that I've seen.
So what about Luke 11:39-51 ?
In Matthew's context this leads into the material from Mark 13. Matthew has Jesus accusing the Pharisees then "forecasts" that all this will come home to them in this generation (the fall of the Temple?), and then we go into Mark 13. It seems to fit Matthew's purposes well. Assuming then that the controversy is something historically plausible that would have separated the followers of the historical Jesus from the Pharisees and the Temple elite, if Matthew composed this, we would have to suppose he was familiar with these issues and could use this to create a plausible controversy in which Jesus could have been involved.
In short Matthew uses a plausible historical fiction to help him make a contemporary point about the fall of the Temple.
I've also proposed that Matthew was very familiar with Aramaic. A short digression one argument that has been raised here against that idea is that Matthew seems to put the last words of Jesus in Hebrew, not Aramaic. However, I think we can see reasons that Matthew, who tells us that not a jot of the law will disappear, might want to have Jesus speak these last words in the language of the OT. I don't think this tells us anything about his competence in Aramaic. Matthew may well have composed things to himself in Aramaic or for a group in Aramaic, before recording them in Greek. He also may have picked things up from an Aramaic speaking group.
But how then do we account for an occasional apparent Aramaic connection between Matthew and Luke? One way would be coincidence. There is a lot of material here, certainly it is reasonable that a small amount would accidentally have some similarity in Aramaic. If these grew more numerous, we might have to look to some form of written Aramaic connection. I've proposed that the author of Matthew claimed to be basing this new gospel on an interview with the disciple Matthew. We need not propose that he went so far as to actually construct this source before hand. He simply composed his gospel and then gave it a providence intended to increase its acceptance. But then suppose our author Luke knows who the author of Matthew is. Luke is setting out to write his new gospel and wants to gather some material. He writes to the author of Matthew, asking for the list of sayings from the disciple Matthew. Now the author of Matthew, with no good reason to refuse, needs to produce this "Aramaic source". He puts together a list and sends it in a letter to Luke. Luke then uses the gospel of Matthew, this Aramaic letter, and an early version of Mark to compose his gospel. (And as has been suggested, maybe he uses some other sources as well). This idea makes me wonder if he had an old copy of Mark on hand, or did he send a letter somewhere in order to get a copy?
In any case, this Aramaic letter would take care of any Aramaic Matthew/Luke connections we might come across, if we decided that coincidence alone was not a sufficient explanation.