Is “mixed math” a useful way to learn math?

Everything you quoted (planetary theory, hydrostatics, mechanics, astronomy, optics, electricity & magnetism, elasticity) in some form or another are still compulsory subjects in maths departments in universities: for example, most academic universities in Russia (MSU, MPTI, NSU, SPbSU to name the top ones), well-reputed Grandes Ecoles in France (Polytechnique, Mines, ENS Ulm,...), etc.

To give a personal example, back in my university on mech-maths department in addition to covering most major fields in maths and mechanics we still had our fair share of physics, quantum mechanics, and CS.

I agree that the proportion of "maths"/"not strictly maths" changed: the sheer amount of graduate-studies-level knowledge in all those fields has increased very significantly since 19th century, so we simply can't ask out students to learn everything to the state-of-the-art level. However, we can ask them to have at least the general idea of other fields, which is still more or less required in major universities.

As for the potential benefit of having the broad view and knowing something from other fields - the ability to draw parallels, to think out of the box is always a plus. I saw some problems in geometry that had very simple interpretation in mechanics; when talking about Lyapunov functions I saw the eyes lighten up after I told the students that "it is some sort of an energy which decreases over time due to some sort of friction", and vice-versa, for some math students the notion of entropy became clear only when we mentioned the Lyapunov function. This list can go on and go on. The most extravagant case I saw was to apply some form of the Heisenberg uncertainty principle in economics for demand prediction.

The basic physics intuition can help while imposing some additional hypotheses on your initial data for some math problem - if one variable has a sense of density, then we want it to stay non-negative. Many maths problem come from the real world - mechanics, chemistry, biology, physics - and therefore we need to understand what we are looking for, what we can suppose and we can not, etc.

So, I would not say that we departed from the "mixed math": on the contrary, it is still there, up to an evolution of our knowledge. And "mixed math" is quite an asset for our skills, ability to draw conclusions, and for our imagination.