Why is Metric TSP's best possible achieved approximation ratio believed to be 4/3? Is it just that integrality gaps (LP/IP) for specific instances do not give more than 4/3? Thanks in priori.
To my knowledge this is the only reason and I am not sure whether everybody in the fields agrees with this. 

The optimistic reasoning is: Since there has not been any progress in getting a larger lower bound, $4/3$ is the right bound. And once we know this for sure, we "only" need a polynomial time rounding algorithm.

More pessimistic people will view the $4/3$ only as a lower bound. To beat this,
you need an algorithm that does not "use" the Held-Karp relaxation, even not in a hidden way. 

Even more pessimistic people would say that $3/2$ is the right bound, because nobody could improve Christofides algorithm (ignoring some of the recent progresses.)

Note that for metric ATSP, the integrality gap is at least $2 - \epsilon$
(Charikar, Goemans, Karloff, FOCS 2004)