In the late 19th century, physics was near the end. Everything related to motion was beautifully summarized in the form of modern mechanics, where all the equations of motion could be derived from the simple, elegant principle of least action. Thermodynamics was finally understood and thanks to statistical physics, explained in terms of the mechanical behavior of constituent particles. All electromagnetic phenomena received an explanation in the form of a set of elegant field equations, which as an added bonus, also explained the behavior of light and optics. In short, apart from some minor, insignificant loose ends, the great project called physics was complete: respected scholars advised talented young students to direct their interests elsewhere, as physics was a dead discipline.
As to those pesky little insignificant loose ends: one was the need to reconcile the constant speed of light predicted by Maxwell's equations with the rules of classical mechanics. The other, more obscure problem related to the nature of blackbody radiation. Very minor issues indeed, with no practical implications whatsoever. Who would have thought that they would lead within a few short years to a complete upheaval of physics and a period of theoretical discoveries that probably has no parallel in the course of human history?
Today, we are actually a lot less close to completing the great project of physics than we thought we were 120 years ago. The Standard Model of particle physics is an amazing achievement, but it has holes in it. Massive neutrinos, the hierarchy problem, indeed the very origin of the up to 26 independent dimensionless parameters that define the theory. And even if these issues are resolved, it is widely believed that the underlying quantum field theory loses its validity at very high energies, and something new is needed. Something new that may be related to another great problem, that of unifying quantum theory with gravity at all energy levels. Cosmology, too, only just became a proper physical science with observational data used to validate theories in the past half century. And it has many issues. Did inflation happen? What about the cosmological constant problem? And so on.
These are not minor loose ends, like the light speed and blackbody radiation issues were believed to be minor loose ends by some in the 1890s. These are major, major issues that will likely require new ways of thinking to be resolved.
It is true that some theories have become very complicated. But the fundamental theories really aren't that complex. A lot of the complication, in my personal opinion, comes from the fact that the way these theories are taught retraces the often convoluted history of their development. Even so, talented young scholars have no trouble absorbing the accumulated knowledge of prior generations of physicists and make meaningful contributions. The fact that in the somewhat blind search for improved understanding, we sometimes end up with theoretical proposals that are truly overly complicated should not mislead us: theories that are inherently complex tend not to prevail, while those that do prevail are soon expressed in simpler form. A perfect example is Maxwell's theory: in its original form it was a horrendous set of 20-some equations, as complicated in appearance as the worst modern theory. This was greatly simplified by Heaviside who wrote down the theory in modern (vector calculus) notation. Then, in the 20th century, the theory was re-expressed in four dimensions and ultimately, using the language of differential forms. Today, I can simply state, "Let A be a thrice differentiable vector field on a Lorentzian 4-manifold and d the exterior derivative operator. Then given F=dA , dF=0 identically. Furthermore, if the manifold is endowed with a metric, I can form the dual G=⋆F and define the current J=⋆dG , which is conserved, as ⋆d⋆J=0 identically." There, that's all there is to electromagnetic theory. Three not terribly long sentences. Doesn't look too complicated, does it?