That is good intuition, but note that one of the general drawbacks of Cramer Rao is that it applies for the unbiased case only (technically no, there is a version for the biased case, I have never seen it used though), and that too only well under certain regularity conditions. Indeed, there is a term for estimators where equality is met in CR, they are called "efficient" estimators.

There is significant research effort these days on going beyond unbiasedness and asymptotics, which in many cases has been beaten to death and is "classical", though there are still plenty of questions.

Here is a simple illustration of unbiasedness/going beyond it. Consider n iid (independent and identically distributed) coin flips of coin of P(heads) = p, and based on the observations, we want to estimate p, call the estimate p'. Optimum estimate in the sense of minimax - min (over estimators) max (over p in [0, 1]) E[(p-p')^2] is not the simple maximum likelihood which is the sample average. Sample average yields a score of (1/4)(1/n), while optimum is (1/4)(1/(sqrt(n)+1)^2), which is slightly lower. It is natural to wonder why people care, typically the motivation is in high dim statistics where there is dependence on the dimension as well. In such a case fine grained characterization of performance sometimes affects the scaling in terms of the target loss amount and dimension, i.e the asymptotics now has at least 2 parameters, sometimes more.

My personal unhappiness with these problem formulations is their lack of robustness to transformations - suppose I now want to estimate p^3 instead of p, the optimum min-max will change in most cases, and definitely does here. Plugging in the sample average and then cubing is far more intuitive, and often works fine even though it is not optimal. There are more extreme examples of this, but that will take the thread too far afield.

If one is interested in the topic of biased estimators, I suggest reading about James-Stein estimators and more generally shrinkage estimators.