There are some problems with the field of statistics. They may stem partially from its role in "certifying" results in other sciences, which induces a certain conservatism, and a desire by many less scrupulous practitioners to just crank out the result they want, never mind understanding how the method works, or whether it is appropriate. It also hasn't (perhaps until recently) been that "sexy" a field, so it may attract fewer really bright people. Finally, there is a tendency for bright people who come into the field from math to think that statistics is a sub-field of math, when in fact there are philosophical issues of inductive inference that are not just math, and practical skills in data analysis that are not necessarily easy just because you're good at math.

Some particular problems:

1) Scary math and pointlessly obscure terminology are indeed a problem. For unnecessarily scary math, the early literature on Dirichlet process mixture models is a good example - almost like they were designed to be incomprehensible to most people who do actually have enough background to use the results. At a lower level, there are pointlessly obscure and misleading terms like "score function", "coefficient of determination", and worst of all, "standard error of the estimate" for the estimated standard deviation of residuals in a regression model (worst since it is not in fact a "standard error" by the general definition of that term).

2) Introductory statistics is generally taught from a naive "frequentist" perspective, because that's been the tradition for the last century or so. The justifications offered in such courses for using p-values and confidence intervals are not defensible - they just sound plausible if you don't know better. There is no good solution, since more sophisticated frequentist arguments will be beyond the level of the course, and shifting to a Bayesian perspective cuts the students off from the scientific literature with p-values, etc. that they will need to be able to read.

3) Outsiders coming into the field often have strange ideas. You might think that physicists capable of building a billion dollar accelerator would be able to recognize when a statistical method they think of is nonsense, but you'd be wrong. There's a tendency for anyone who learns information theory before statistics to think that information theory is tremendously relevant - but no, rephrasing maximum likelihood or Bayesian methods in information theory terms may sometimes be slightly helpful in thinking about them, but doesn't really add anything fundamental. And no, there's nothing particularly special or interesting about distributions that maximize entropy subject to some (generally arbitrarily selected) constraint.

4) There's a tendency to want more than you can get. There is no one "objectively correct" model/prior/analysis for a data set. Subjective assessments are unavoidable. But a lot of people don't want to accept this fact, and devote great efforts to ways of trying to pretend otherwise.

However, if you think statistics is just a simple matter of running a curve through a cloud of points, you're very wrong. Even running a curve through a could of points is a complicated and subtle enough task that deep issues arise, and these issues become much more obvious if you're trying to fit a function of hundreds of variables rather than just one. And if you're trying to not just "fit" data but come to valid conclusions about cause and effect, or about underlying latent variables that will provide useful information in new contexts, then you really do need to know a lot.