I guess the cultural significance likely isn't too massive and I suppose that at first glance the date might not even stand out. I would put money though that more Brits know what happened on the 4th of July though than say the 16th of August. As I mentioned in another comment the events of American Independence happened far earlier than any of the other colonies and under massively different circumstances which were much less voluntary on the British side. How many of the other colonies successfully gained independence through a war with the British empire at the peak of its power rather than due to treaties passed in an era with massive public sentiment against imperialism.

Economically, militarily, and politically...yeah..

I will concede the point that there are a lot of days and it would be impractical to avoid all of them for elections. However, there is a pretty massive outlier. Most of those countries that were colonies gained independence after WW2 (Iraq and Afghanistan being notable exceptions) due to a strong shift in public sentiment against imperialism. The US on the other hand gained independence 200 years before any of the other countries on that list in a war between colonists and what at the time was one of the most powerful empires in the world. Culturally speaking I'm sure that even if they don't care too much, British citizens are much more aware of the history of American Independence than most of the other countries on the list since it was a much more dramatic affair. I will say Irish independence would likely be more relevant to British citizens and come with a higher level of historical awareness and even emotional attachment.

EDIT: I will add that the US also has massive cultural significance on a global scale. How many people in the UK watch shows, movies, and listen to music from Egypt compared to from the states? If you look at the music charts in the UK they're dominated by American artists.