Loudspeakers have we really made that much progress since the 1930s?

In terms of new physics or fundamental technologies (as someone pointed out, planar was also from the 1930's), there is nothing new since then for loudspeakers. We are still pushing air to make soundwaves and the method of generating air motion is down to moving diaphragms/membranes/cones.
Where there has been new technology advances is in electronics i.e. amplification via solid state devices and digital technology (both for amplification and source recording/playback). The advances in electronics allow for much greater power to be send to speakers which have allowed for smaller speakers that can more readily fit into modern living space and even automobiles to generate the same or greater SPL (sound pressure level) as large horn speakers of 1930's. Solid state and digital technologies have allowed for miniaturization of components and also reduce costs and ease of mass production to enable the mass public to enjoy high fidelity music. This is where there has been advances. In the 1930's, high fidelity music was limited to the wealthy and the few. 
I myself enjoy low powered SET amps and single driver high efficiency loudspeakers and I do enjoy a good horn speaker and generally prefer them to modern speakers. And in this regard, I don't feel that modern, contemporary loudspeaker systems sound better than the vintage ones that have been setup appropriately ... at least to my ears. FWIW and IMHO.

But it is clear that speaker systems
for concert performances have improved dramatically from the experiences I had in the Boston Garden in the early 70s. Sometimes there was so much distortion it greatly detracted from the enjoyment. Now we mostly hear very clearly at high volumes.

Yes and no.  It would be very difficult to find whole speaker systems, and even modern speakers using vintage drivers, that would make compact monitors that sound as good as modern monitor speakers.  The same probably holds true for thin column-type floorstanding speakers.

But, for systems where one is not constrained by practical considerations, such as size and ugliness, I would go with a modern system built with vintage or replica vintage drivers over anything I've heard from modern speakers.  If size and cost is not an issue, something with Jensen M-18 fieldcoil woofers and Western Electric 555 drivers and 15A horn and 597 tweeter would be nice to have (and a room big enough for such a system would be really nice).  For a "compact" system, something with the Jensen/ERPI M-13 fieldcoil and a 597 tweeter would also be a nice choice.  I also happen to like my system which is built around the Western Electric 713b midrange compression driver (made sometime around 1939).

At the Capital Audiofest, held this past weekend, Deja Vu Audio was showing a recently-built speaker featuring vintage drivers.  The speakers had 15" Jensen woofers and midrange compression drivers from Yoshimura Labs (a 1960-70's Japanese company that made theater/public address systems that emulated Western Electric drivers) in an open baffle configuration (it looked like a box with a closed back, but the back allowed sound to pass through).  It is hard to beat this type of system for delivering harmonic "denseness" (rich, saturated sound) and a relaxed (not edgy) and natural presentation while also delivering terrific dynamics, speed and clarity.

The BIG plus with many vintage systems is that they can be run with a wide array of low-powered amps.  To me, the best sound is delivered by lower powered amps, regardless of whether one goes with tubes or solid state.

As usual LarryI has a balanced and measured post which means I agree with him.  There is no question that modern speaker design software and measurement tools have contributed to an overall improvement in speaker sound quality.  For those of us who were kicking tires in hifi stores in the 1960s and each decade since, I can say that there are fewer poor sounding speakers today than in years past.  This is especially true with lower price speakers and smaller speakers. 

But if we ignore price and size considerations, the very best vintage drivers and horns have some special musical qualities that are not equaled by today's speakers at any price.  I have only limited experience with the big Western Electric horns (and no experience at all with the Shearer horns mentioned by the OP) but each time I have heard them left no doubt they are capable of excellent sound quality.  On a much more accessible level, I find that my own speakers using a vintage 15" Jensen paper cone woofer and Altec 32A horn with 802-8G compression driver (dating probably from the 1970s) are very satisfying musically. 

Sal,
Yes, Larryi comments are consistently insightful and well-thought-out.  Larryi in general I agree with you preference for the lower power amplifiers in regard to superior sound quality. This point is driven home often when I hear the mega power amplifiers driving difficult load/inefficient speakers. 
Charles,