Are these ‘GGT-less primes’ already known?

I was playing around with Goldbach representations and came up with the following class of primes.

1. Consider an even integer $n$ that can be written as a sum of two odd primes,
   $$ n=p+q,\qquad p\le q. $$

2. For a fixed $n$, define $GGT(n)$ to be the largest possible value of $q$ among all such pairs $(p,q)$.

3. Since the case $q=n-3$ is trivial whenever $n-3$ is prime, exclude all such $n$. Let $n'$ denote the remaining even integers.

4. Call an odd prime $q$ a GGT prime if $q=GGT(n')$ for at least one such $n'$, and a GGT-less prime if this never happens.

Computing the GGT-less primes gives

$$ 5,11,17,29,41,59,71,101,107,149,\ldots $$

This sequence looks very similar to the sequence of smaller members of twin-prime pairs, apart from $3$. In fact, every GGT-less prime is the smaller member of a twin-prime pair.

To see this, let $q$ be a GGT-less prime. Consider $q+5$. The pair

$$ (5,q) $$

is one Goldbach representation of $q+5$. For $q$ not to be $GGT(q+5)$, there must be another Goldbach representation whose larger prime exceeds $q$. Since the smaller prime must then be less than $5$, the only possible odd prime is $3$. Thus the only possibility is

$$ q+5=3+(q+2). $$

Therefore $q+2$ must also be prime. Hence $q$ and $q+2$ form a twin-prime pair, and $q$ is its smaller member.

The converse, however, is false. The first counterexample is $137$. Although $137$ and $139$ are twin primes,

$$ 148=11+137. $$

Any Goldbach representation of $148$ with a larger prime than $137$ would have to use an odd prime less than $11$. The only possibilities for the smaller prime are therefore $3,5,7$, giving

$$ 3+145,\qquad 5+143,\qquad 7+141, $$

none of which is a Goldbach representation. Hence

$$ GGT(148)=137, $$

so $137$ is the smaller member of a twin-prime pair but is not GGT-less.

I searched OEIS for

$$ 5,11,17,29,41,59,71,101,107,149 $$

and did not find a matching sequence. A closely related sequence is OEIS A178128: after omitting the initial term $5$, the first nine terms agree, although the two sequences eventually diverge.

Is this class of GGT-less primes already known under another definition? If not, is there any heuristic or known result suggesting whether infinitely many GGT-less primes should exist?