Limitations and heuristics on a twin prime generating algorithm

In the rather new MSE post Recursive sequence $a_{k+1} = a_k - \gcd(a_k, (n+k)^2 - 1)$ generating twin primes, a sequence of integers $(a_k)_{k \geq 1}$ is associated to each natural number $n$, namely $a_0 = n^2$ and $a_{k+1} = a_k - \gcd(a_k,(n+k)^2-1)$.

What is interesting, as pointed out by the user, is that by defining $T(n)$ to be the infimum $m$ such that $a_m = 0$, for which we have the bound $T(n) \leq n^2-1$, numerically the pairs $(n+T(n)-1, n+T(n)+1)$ seem to always be twin primes.

Does there exist any reason why this should hold true in general, other than the numerical checks up to $n = 2,000,000,000?$ The author shows that if in the sequence, the term $I_k = a_k + (n+k)$ is ever between two twin primes then $T(n) + n$ is always between two twin primes. Does there seem to exist a structural reason (heuristical, otherwise this would be a proof of the twin prime conecture) for the sequence to produce twin primes, or is it simply a case of the law of small numbers, whereas for $n \ll +\infty$, the density of twin primes is heuristically much larger, hence the larger the chance that some $I_k$ is going to be between twin primes.

In general, how large of a contribution would this sequence be to modern mathematics, specifically in the problem of prime gaps.