Heuristically this should be the case. For any prime p greater than 5, consider the set of numbers of the form $2^a3^b5^c p \pm 1$. The "probability" that one of of these is prime should be about $$\frac{1}{\log (2^a 3^b 5^c)} = \frac{1}{a \log 2 + b \log 3 + c \log 5} .$$ So the probability that both $2^a3^b5^c p + 1$ and $2^a3^b5^c p - 1$ are prime should be about $$\frac{1}{\left(a \log 2 + b \log 3 + c \log 5\right)^2}.$$ Now, note that the series $$\sum_{a,b,c} \frac{1}{\left(a \log 2 + b \log 3 + c \log 5\right)^2}$$ diverges, so we should expect for a given $p$ there should be such an $a$,$b$ and $c$. So after noting the twin prime pairs $(3,5)$, $(5,7)$ and $(29,31)$ it seems like we should expect a much stronger statement. For any prime $p$, there should be a positive integer $n$ such that $p$ is the largest prime divisor of $n$ and $n+1$ and $n-1$ are both prime.
Obviously, proving something like this is well beyond current technology. I'd also say that it is highly likely that even your weaker statement is well beyond what is currently doable.