That's one part of the game. A second part is that a thinner core has a lower value for the longitudinal elastic modulus which results in a softer feeling when playing the string. This of course is helpful, when you play on a string with an increased level of tension.
A third part is that modern string models mostly have an increased level of tension. The higher you choose the tension of a stretched string, the higher the value for the wave resistance (that's the mechanical equivalent to the impedance of an electric circuit) of this string will be. This results in a stronger damping of the harmonic vibrations of the string. There is no way to compensate for this other than reducing the string tension. But a small improvement can be achieved by reducing the bending stiffness of the string, which of course is done by reducing the diameter of all components the string is made of.
That is why thinner stings are indeed intentional when producing strings of high tension, and the use of materials with a higher density is necessary to be able to produce strings with less bending stiffness. Mostly to avoid that the sound of this higher tension strings gets too "focused" and would be more or less become comparable to what you get from a sine-wave-generator.
"Would the stress or tension per square inch be larger on the thinner string in some instances?"
To understand the answer, it has to be clear that the tension of a string is mostly carried by the core of the string.
The answer to the question is: yes, that can happen in the following cases:
i.) When you keep the cross-sectional area of the string core constant, the stress increases proportional to the tension. That case occurs by increasing the weight of the winding of a string without changing the string core, the vibrating string length and the pitch.
ii.) When you keep the tension constant, the stress increases inversely proportional with a decreasing cross-section. That case can be accomplished by decreasing the cross-section of the string and compensating the resulting loss of mass of the core by adding that same mass to the winding of the string. Again the vibrating string length and the pitch are constant.
"Would a thinner string always be softer than the thicker versions of the same string?"
No, this is not a general rule. If a thinner string will be softer depends on the materials the compared strings are made of and on the relations between the geometrical and physical parameters of the materials the both strings are made of.
I hope this answers your questions.
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The lightest metals in common use for that purpose might be aluminum, copper, titanium and iron/steel. The heaviest are likely tungsten, platinum and gold.
So by using a heavier (i.e., denser) metal a string of the same pitch can be made thinner, to keep the weight the same; Sometimes too thin for some of us. Or. alternatively, the greater mass of the string can be compensated for by tuning to a higher tension, which can also make it more uncomfortable for the players' fingers and change the response of the instrument.