Perfect sampling is now called exact sampling and it is indeed exact, when compared with a standard MCMC algorithm that is only asymptotically exact (in the number of simulations). An exact sampling algorithm comes with a stopping rule that guarantees that the final value is simulated from the target distribution. Although it may require a longer execution time than an inverse cdf method, a known transform of a simpler distribution (e.g., a Negative Binomial being the mixture of a Poisson distribution by a Gamma distribution), an accept-reject or a ratio-of-uniform algorithm, it identically operates as a function that returns one or $n$ simulations when provided with a seed.
Concerning the grid inversion method suggested in the question, this is an approximation of an exact inversion, with cost increasing with the requested precision. In addition, it does not extend to larger dimensions.
Remark 1 The compactness restriction is not a major issue. First, the random variable $X$ to be simulated can be transformed by aan arbitrary function $H$ so that $H(X)\in(0,1)$$H(x)\in(0,1)$ for all realisations $x$ of $X$. Second, the bounds $a$ and $b$ can be chosen so that $\mathbb P(X\in(a,b))$ is (at least) equal to a predefined probability like $0.99$ or $0.9999$.
Remark 2 There exist other "perfect" sampling strategies, including accept-reject, ratio-of-uniform algorithms.