Author here. The table-maker's dilemma is a problem if you insist on correct rounding. However, interval arithmetic does not require correct rounding. It merely requires rigorous error bounds. It turns out that a lot of the time when people ask how to obtain correctly rounded results, they are actually asking the wrong question, and what they really need is just error bounds.

This is why Arb has a lot more functions than MPFR: it's extremely difficult to implement correctly rounded functions as required by MPFR; Arb's contract (returning an error bound) is far easier to satisfy and works just as well 99% of the time.

From a software development point of view, the problem with correct rounding is that it doesn't compose: given correctly rounded functions f and g and correctly rounded input x, f(g(x)) is not (in general) correctly rounded. In interval arithmetic, the corresponding composition law (the inclusion property) does hold; this means that you can compose arbitrarily complicated functions without thinking.