It was common because a physics degree is essentially applied maths, and physics PhDs will have had to solve research problems nobody has solved before. Sell side has had less of a need for such a generalist degree since the advent of MFEs and the like, and buy side is now increasingly hiring computer scientists and now that data science and machine learning has taken off as its own branch of, if not science, then at least university esucational prgrammes, they, not physicists, are en vogue on Wall Street.
Physicists still get hired given the history of hiring them, and so some hiring managers may still give them a break and universities from their ivory towers keep saying that an unsuccessful academic in physics can always turn to finance, but this is increasingly less true and physics majors should know this.
As for statistical mechanics, it does not cover the necessary tools: there is for example no time series analysis done on courses because it is not really of interest in carrying out the theory of ensemble averages, and thermal equilibrium, and the approach to probability is very different: The theories and models in physics are much more correct than those in finance, so there is no reason to, for example, study vanilla models separately from term structure models. This means physicists will not learn the fundamental theorems of probability useful in finance, like changes of measure and all the related stuff, nor would a physicist be so averse to a Monte Carlo solution as you don't need to be calibrating on the fly to some targets out of simpler models.
Instead, physicists solve Ornstein-Uhlenbeck analytically, but the problems of interest in statistical mechanics beyond ones with easy solutions, are, for example, stochastic PDEs, which you'll formulate and worry about the Ito-Stratonovich dilemma, but in the end just write up a numerical solver for. It is also the case that in finance the solution is an expected value, whereas in physics one often is interested in for example the dynamics of the probability density function itself: Not much here you can simplify with martingales, I don't think.
