I did not get how "the sum of three primes" can help with the search for prime sextuplets ?
Also HardyLittlewood circle method is a technique used to prove asymptotic behavior of a series. AFAIK there is no generating function for prime numbers etc.
Video is from 2014 with bad sound (and no sound until 2:50), lost patience soon.
There may be ideas to use, only I may not see them.
I was referring to another work from the same author. The same man who worked on the "the sum of three primes" problem has recently been working on a method for making sieves that uses less RAM: that's the work that may (or may not) be helpful, but I still couldn't find any links to an actual paper.