I'm afraid you are advancing from a coupled flawed suppositions. To begin with most democracies (including the United States) did not start with "first-past-the-post" voting. In the countries in which this form of democracy is present, it is often the result of political evolution.

For example in England, Parliament has a nearly one thousand year old history. During most of that time it only vaguely looked like its current incarnation. It was a long path from the early parliaments dominated by powerful lords to the modern republican House of Commons dominated parliament.

In the United States at our founding, we used the first-past-the-post model only for one branch of Congress, the House of Representatives. The Senate was elected indirectly by the state legislatures of the respective states.

Presidential elections were also dramatically different affairs in the early republic. To begin with in many states there were no presidential elections as such, instead state legislatures assigned the state's votes in the electoral college. There was also a system set up to send the Presidency to the House of Representatives in the event of a lack of a majority (which happened only two or three times depending on how you are counting 1876), which many people believed would be the most common outcome in such a big country, before the advent of nationwide political parties.

As you can see the framers of the Constitution actually favored a number of indirect elections (which could have multiple ballots), to straight up first-past-the-post democracy. It was only with a number of reforms over centuries, that America gained her modern electoral system.

Your second supposition is that First Past the Post is universally agreed to be the "worst system." While reasonable people can (and should) disagree about the advantages of different systems, I think this claim strays into hyperbole. There are many advantages to FTTP (and the two party system which it reinforces), which are sometimes brushed aside in these discussions. FTTP creates geographic responsibility, promotes stability and allows politicians independence from their political party.

A better way to ask this question might be: "How did <country X> end up with its modern electoral system?"

That is likely to get you better historical answers. You could also try asking in one of the social studies subs about the general development of democracies, and see if you can find a political scientist to discuss commonalities in the development of democracy.

Could you please edit the claim that FPTP "near universally agreed as the worst voting system" out of your post? It's not particularly relevant to your question and is basically asking for a political debate, which isn't what this subreddit is about.

You've answered your own question, in a way. The modern concept of Democracy owes largely to the Westminster System established in the UK. Most commonwealth and ex-commonwealth countries (the United States, I don't think, is technically considered an ex-commonwealth states but is the most famous Democratic system) are based on the Westminster System by virtue of having been previously governed by Britain. This is how their own independent legislatures were set up.

Many Western democracies are not FPTP but are representational in different ways, like those of Germany and the Scandinavian countries.

As to your judgement of the Westminster Model being the "worst", again you've answered your own question. It hasn't changed much because it seems to be working pretty well for the UK and the other commonwealth/ex-Commonwealth countries. To change the system would require a dramatic overhaul and as a Canadian who hears about the suggestion on a fairly regular basis, I don't think there's really that much of an appetite for fixing what isn't broken. Yes, aspects of our Democracy can be improved but there are always trade-offs.

Here's an article on the mechanics of how the Westminster system was adopted elsewhere.

A quick glance at the Wikipedia article on FPTP shows that it's probably not as common as you think, and most European democracies other than Britain don't use it.