Variations in pitch and tuning systems stem all the way back to Renaissance / pre-Renaissance. Before a standard tuning system was codified, musicians in each city basically had their own tuning system, and instrument makers were limited with what they could do. The result was that if musicians from different towns / cities met, they wouldn't be able to play with one another as their instruments were not in tune.
Once technology increase, tuning systems were able to be consistent across countries - thereby improving consistency of sound and instrument creation.
To this day, pitch still continues to vary, though it is purely by choice relating to preferred aesthetics. For example, here in the US, "A" is tuned to 440hz, while in Europe, it may be 442 or higher. The reason for this is exactly the question you posed in your post - that Europeans desire a slightly more brilliant sound. Pitch inflation itself cannot be attributed to an ensemble's brilliant sound, but it can be attributed to an ensemble's desire for a more brilliant sound.
So, how does it work?
It is the same reason why audio engineers boost higher frequencies on cymbals for the extra bright, crisp sound. It works through the way sound waves resonate.
I will omit all the figures and pictures that would make this explanation easier, and unfortunately only provide a very brief overview for sake of time.
Think about the overtone series. On a string, touching the string at specific nodes will generate different divisions of the wave that still divide evenly into the original vibration of that string. Let us say that you now have a second string, tuned slightly higher, but for all intents and purposes is perfectly in tune with itself. Played by itself, it will still yield all of the same overtones from the same string divisions, however, when compared to the original string, the sound waves of each string resonating openly will not align. Since the second string is tuned slightly higher, its sound waves will be sounding ever so slightly faster than the original string. This discrepancy between strings causes the second string to stand out over the first, creating what most musicians consider to be a more "brilliant" sound.
While I have never heard of individual musicians doing it, it is certainly possible for some orchestras. In the US, in a professional orchestra, they are looking for as homogenous a sound as possible. An individual performer preferring to not tune with the group would not have a job very long. Generally, an orchestra will decide what the entire group will tune to, and in my experience it has little to do with "out doing" another musician, and more to do with achieving preferred aesthetics. After all, there are many other ways to better "out do" another group.
Lastly, I must correct the quoted paragraph in the OP and say that tuning a string higher does not objectively give more amplitude to higher overtones. Technically, since the string tuning is changed, it actually gives you a completely different overtone series. However, since that series so closely relates to the original series, it gives the impression of amplification. Again, recording engineers use this technique and others to create a more "full" sound.
Hope that helps.