Interesting thread. I heard a different interview with him on the BBC. https://www.bbc.co.uk/programmes/w3cswllf
I was actually thinking about his accent when I was listening to that interview, before I saw this thread.
I think his accent is pretty good. His goal was to live in New York, where lots of people have accents, so he wouldn't stand out as much there. Probably more likely to trip him up would be an insufficient knowledge of American culture. Not the art and history he would pick up at the museums he went to, but the radio programs, movies, music and cartoons that "everyone" his age would have been exposed to growing up in the US.
In fact, literature with three syllables sounds so BE. I can hardly say it that way and keep a straight face. It's one of the things that just sounds so exaggeratedly BE, like aluminium with 5 syllables.
As for the German mother explanation, it could work for most people. While most children of immigrants who grow up and attend school will speak perfect AE, not every last one does. I know two Americans of Japanese heritage. The son, who is about my age, was born in Japan, but came here as a toddler. He attended American schools and college. He speaks English fluently, but he makes the odd semantic or grammatical mistake and he has an identifiable accent. His sister, younger by a few years and born in the US, speaks semantically and grammatically perfect AE, but she has a slight identifiable accent. I've never been able to quite figure it out, but I would say there is something Japanese about the prosody when she speaks.
Two other examples from Spanish backgrounds. We have a good friend, born in Colombia, who immigrated with his parents at two. They lived in New York and never learned English. He was an only child (i.e. no siblings to speak English with at home). He attended school in New York and went to college in California and Michigan. He speaks otherwise flawless English (grammar, vocabulary are excellent) with slight prosodic quirks.
A former colleague of mine was born in southern Texas, the child of immigrant farm workers. She came north to Michigan to attend college. She also speaks flawless English with a prosodic hint of Spanish. I gather, that she spoke a lot of Spanish growing up in Texas, even at school.
I think whether or not a child will develop truly native capabilities and a flawless accent in the country's dominant language depends on how much time the children spend in their first language and culture and the extent and kind of contact to the second language.
What I found myself wondering about Barsky was how much his English has improved. Okay, it's quite good now, but was it that could when he first arrived? It seems unlikely. But as Qual der Wal pointed out, he managed to survive for many years here without running afoul of spycatchers.