The short answer is that we are inside the Milky Way instead of looking at it from outside.
And it is not exactly true that it was not until the 1950s that the shape of the Milky Way was figured out.
As soon as Galilleo pointed a telescope at the Milky Way and resolved it into many stars, it has been known that the light of the Milky Way is composed of the light of many, many stars blended together.
There were three possible asumptions about the Milky Way stars.
One) that they were smaller than the naked eye stars, and so could be at the same distance as the naked eye stars.
Two) that they were so much smaller than the naked eye stars that they were much closer to Earth than the naked eye stars.
Three) that they were as large as the naked eye stars,and thus were so much fainter than the naked eye stars because they were very much father away than the naked eye stars.
Starting in the 17th century astronomers tried to measure the parallaxes of stars to find their distance. In the Hellenistic age, it was realized that if the Earth revolved around the Sun, the positions of the stars should move during the year as the Earth moved. Since no such movement could be detected in ancient times, that was one of the arguments used against the ancient heliocentric theory.
When early modern astronomers accepted the Heliocentry theory, they realized that detecting the parallax of stars would be a major piece of evidence for the heliocentric theory and would help figure out the size and shape of the universe.
So attempts to measure the parallaxes of stars were made and failed for over 150 years. If any astrnomers believed that the Milky Way stars were closer to Earth and chose to measure their parallaxes, the failure to do so would prove that the Milky Way stars must not be any closer than the naked eye stars. Finally, more advanced instruments in the 1830s enabled the parallaxes of 61 Cygni, Alpha centauri, and Vega, to be measured, followed by those of a few other stars in the following decades.
Those proved, among other things, that stars were not all at the same distance, so they were not all on the surface of a spherical shell around Earth. They also showed that, when the apparent magnitudes of stars were compared to their distances, the absolute magnitudes of stars could be calaculated, and those absolute magnitudes varied greatly. So the apparent magnitude of a star was a very poor indicator of the distance of that star.
Back in the 18th century, the Milky Way was assumed to be either a disc or a torus shape with the Sun and Earth inside it. Thus the stars of the Milky Way appeared as a band across the entire sky as seen from Earth, inside the Milky Way.
By then number of tiny hazy patches of light - "nebulae" - in the sky have been discovered, and some people speculated that since they looks like tiny bits of the Milky Way, the Milky Way might not be the entire universe, and there could be other things like the Milky Way at great distances from it.
Proof of the Milky Way consisting of many stars came in 1610 when Galileo Galilei used a telescope to study the Milky Way and discovered that it is composed of a huge number of faint stars. In a treatise in 1755, Immanuel Kant, drawing on earlier work by Thomas Wright, speculated (correctly) that the Milky Way might be a rotating body of a huge number of stars, held together by gravitational forces akin to the Solar System but on much larger scales. The resulting disk of stars would be seen as a band on the sky from our perspective inside the disk. Wright and Kant also conjectured that some of the nebulae visible in the night sky might be separate "galaxies" themselves, similar to our own. Kant referred to both the Milky Way and the "extragalactic nebulae" as "island universes", a term still current up to the 1930s.
I note that the German astronomer Johann Heinrich von Maedler (1794-1874) believed that all the stars in the Milky Way revolved around a center of mass (correct), and that there was a central Sun at that point (not so correct), and that point was at or near Alcyone in the Pleaides star cluster (which much too close to Earth and in the opposite direction from the real galactic center).
My copy of Men of Other Planets by Kenneth Heuer, 1951, 1954, says on page 134:
And there are estimated to be upwards of 10,000,000,000 stars alone in the great system of stars we call the Milky Way.
Furthermore, the sun has not received any special privilege, although it is suspected to be larger and brighter than the average star. Comparing the shape of the Milky Way Galaxy to a wheel,the sun is located about 3/5 of the way from the hub to the edge, while the entire star system has a diameeter of approximatley 100,000 light years.
And there is a rough diagram of the shape of the Milky Way on page 134.
So by 1951, or 1954 at the very latest, the shape of the Milky Way Galaxy was known fairly accurately by contempoarary standards.
The first project to describe the shape of the Milky Way and the position of the Sun was undertaken by William Herschel in 1785 by counting the number of stars in different regions of the sky. He produced a diagram of the shape of the galaxy with the Solar System close to the center. Using a refined approach, Kapteyn in 1920 arrived at the picture of a small (diameter about 15 kiloparsecs) ellipsoid galaxy with the Sun close to the center. A different method by Harlow Shapley based on the cataloguing of globular clusters led to a radically different picture: a flat disk with diameter approximately 70 kiloparsecs and the Sun far from the center. Both analyses failed to take into account the absorption of light by interstellar dust present in the galactic plane, but after Robert Julius Trumpler quantified this effect in 1930 by studying open clusters, the present picture of our host galaxy, the Milky Way, emerged.
So this shows that the shape and size of the galaxy, and the Sun's position in it, was determined by astroomers sometime during the period of 1930 to 1951.
I note that with the developement of radio astronomy after WWII, astronomers traced the radio waves emitted by interstellar clouds of hygrogen and began to use them to trace the spiral arms of the Milky Way. This was already being done by the 1960s.