My impression is that historically national flags around the globe were much more varied, but over time they’ve shifted towards simple tricolours.
Is my impression correct and if so why has this happened?
While there are some examples of tricolour flags before the 18th century - most notably the Prince's Flag of Netherlands - it was the French Revolution that was the watershed moment for this design.
At the time of the storming of the Bastille in July 1789, members of the Paris militia began wearing blue and red cockades on their hats, based on the traditional colours of the city. By end of July, the Marquis de Lafayette - who had played a central role in the Estates-General and the drafting of the Declaration of the Rights of Man - introduced a new national flag which combined the colours of the cockade with the royal white: the famous "tricolore". This flag was formally adopted by the Constituent Assembly on October 24, 1790.
This flag had resonance not only because it symbolized a "balanced" constitution in which royal power was checked by a government elected by the people (at the same time the cry "Vive le roi!" was replaced by "Vive la Nation, la loi et le roi!", or "Long live the nation, the law, and the King!"), but also because its simplicity reflected a faith that the Revolution could remake society through the principles of reason.
The tricolore was removed as a French national symbol with the restoration of the Bourbon dynasty in 1815, and replaced once again by the royal white. But it remained a symbol of revolutionary republican ideals, and after the July Revolution of 1830, Louis-Philippe - who presented himself as a "citizen King" - restored the tricolore as the national flag.
In the ensuing decades, and particularly during the Revolutions of 1848, the flag became a symbol of republican, liberal, and nationalist ideals - not only for France, but for other European countries. So, for example, liberal reformers in Germany by the 1830s adopted a tricolour flag with the colours of the Holy Roman Empire (black, red, gold) as their national symbol. In other countries, the tricolour combination of green-white-red became the symbol of republicanism and was subsequently incorporated into the flags of Hungary, Italy and Mexico.
As various other young republics in Europe adopted flags, the tricolour became the model of choice because it symbolized a commitment to the liberal and democratic ideals of the French Revolution and the Revolutions of 1848: Irish and Romanian nationalists adopted the design in 1848, as did Belgians (although Belgium had already developed a tricolour design during the Napoleonic wars).
In the long run, it may be that other countries have adopted tricolour flags because of the simplicity of their design, but in the 19th and early 20th centuries, the symbolism of the design was closely connected to the ideals of republicanism, liberalism and democracy that were first espoused during the French Revolution.