I've always understood "the largest known primes" to colloquially refer to primes which (i) have decimal expansions which can be written down in a short amount of time and (ii) have primality proofs which have been double/triple checked.
I imagine that if the sequence you suggest were shown to always be prime, people would stop talking about the largest known primes. They might continue to search for ways to find lots of primes of the same large size, and improve the bound on that size.
