And that part is the key. There’s nothing magical about 440 cycles per second. But if a luthier crafted an instrument to produce the best resonance at that precise frequency (and its harmonics), it can make a huge difference.
In fact, this is where the trend in the 1970s to use instruments as they were originally planned may make the most sense. It might be futile to try to play the violin in exactly the same way it was played in Stradivari’s time. But a violin made before 440 Hz was adopted as a standard probably doesn’t sound as good in a contemporary ensemble enforcing the 440 tuning as it would in a tuning emphasizing its natural modes. Not to mention the dominance of 12TET in almost any musical genre since the middle of the Twentieth Century, including in the most diverse of the World’s musical traditions… Using a twelve-tone equal temperament based on A440 to play an instrument meant for Just Intonation at another reference pitch is like using a hammer to hit a marimba. It may work, but it doesn’t sound so good and it won’t help the instrument last very long.