There surely are good reasons to pick another model rather than the US or perhaps mix various models if one is determined to build the "best" democracy / society as other answers have pointed out.
There is one main issue however aside from that: A country needs a unified will to change its own model and needs to agree in the direction.
A country that works well won't see the need to (drastically) change its model.
A country that is dysfunctional, dictatorial or anti-democratic will typically not easily find the unified will to change itself into a model democracy. A dictator in power would gain nothing from it. There might be an opposition that claims it wants a democracy, but often a good portion of opposition forces in essence simply wants to replace the dictator but keep an authoritarian approach - simply because that is the mindset they grew up with. If there truly is a pro-democracy revolution, changes implemented are often a messy mix of democratic principles the former opposition leaders felt are important and traditions they felt were too unpopular to oppose.
Then again, most countries do change their laws and constitution on a regular basis in an incremental fashion. Who is to say some of the leaders of those countries don't look to the US for inspiration. Typically changes are however more driven by immediate needs than an attempt to mimic a particular country. Note that political systems are strongly connected to other aspects of a society and what works well in one country and will be accepted there won't be easily accepted or have the same beneficial impact (on its own) at least short term than in other countries.
If we look at the US, there are many archaic aspects to its political systems and aspects that feel extreme or reckless to other countries, examples:
- the electoral college seems mainly in place for historic practical reasons not as a particularly good democratic mechanism
- gun control and health care are at odds to what Europeans would think sane, but perhaps would be closer to a developing nation such that it could find easier acceptance there
One aspect in particular is that the US is build on a federal model - which makes sense for its size and history, but not for a country like, say Iceland.
So a 1-1 application typically doesn't make sense. When designing a law framework you always need to account for local history, tradition and societal norms. Some of these you can shape over time, some of them you cannot (or only over a really long time without direct influence).
All that aside, since the introduction of certain models - especially in a democracy or if you want to establish one - hinges on the support of the masses, it is far more important to change and influence general values and knowledge than build a particular framework of law. If you want to change a country completely, you first need power and stability, then education and people getting accustomed to a change into a democratic direction. So you need to introduce changes in the law framework in a way they get accepted and in coordination with changing attitudes in your country. The details of the democratic system are far less important than the conviction to uphold democratic principles throughout the society. And the US is not a particularly good example framework to get there. Education and will for self-determination was already strongly ingrained in at least the European settlers that made up much of the original US citizenry, as they were European educated and on top typically the ones most unsatisfied with their European governments - thus arguably the most unruly ones, most striving for self-determination (speaking broadly again). One can at least question whether the current US society, education system and law framework is the best to inspire democratic convictions as it was shaped with an already democratic/self-deterministic mindset implanted in the populace and has its own pragmatic aspects rooted in its particular history (e.g. "you need to defend yourself in the wild west").
All that being said, there are countries who radically change and try to emulate other models. But this will typically be about picking the best pieces and melting them into a pot together with local specialities. Consider Germany or Turkey. In Germany after WW2 there was likely the closest a country can come to a clean slate start as the main supporting force of the old model was beaten to death (literally) to at least a large degree. And while many democratic and federal principles were incorporated under outside influence that wanted to ensure a hostile dictatorship does not re-emerge easily, the result isn't a copy of an existing constitution / framework of law. It is still very much a unique framework based on the German history.
In Turkey, Atatürk did try to emulate a lot of Western principles and managed to change his country in large parts into a secular model society westernising much of its make up. However, his goal was also not to emulate a particular country but to improve his home country and pick the parts from Western society he considered beneficial (experts in that matter feel free to correct me^^). Turkey is also a good example that change takes time to permeate society, if it does so at all. Today Turkey gradually slips back under leaders that have other goals. Perhaps goals in the opposite direction, if they care about shaping their country in this top-level sense at all (again not an expert in temporary Turkish politics^^). At least they challenge some of the protections placed into the law framework and some would argue that exactly the democratic protections prevent the country's protectors from stepping in (e.g. a military coup from a military that would try to uphold Atatürk's principles is still a coup).
Finally, another practical reason why people don't exactly want to emulate the US system:
Publicly declaring the US as the model (or a particular other country) can be very unpopular. First, implying your country is inferior compared to a particular other country will typically not go well with a nationalistic voter base. And if you want to change your country substantially, you might need at least some form of patriotic/nationalistic sentiment. While this issue can be overcome, it's just a drawback that you will think twice before you accept it.
Second, the US in particular is not as loved around the globe as it would like to be - especially by the population of many countries that are not so well off. South America has a history of US influence, that a good part of the countries don't particular appreciate. Many countries in the middle east have policies and convictions at odds with the US, have been in wars that at least some blame on the US or have had the US as occupying force. Close allies to the US on the other hand already have some form of democracy and are well off typically on their own (exceptions may apply). So publicly declaring to aim to copy the US in particular would open an easy way to distract from the goal at hand for any opposing political force.
The other big general reason is exactly the shape in which these countries you mention are. They are typically plagued with corruption and a history of dictators taking power from one another. Struggling democracies often have small scale populist "dictators" that misuse the system. If there are shining figures to right the wrong, the general sentiment will often be that they need all the power they can get to fight the corruption. So any means to introduce democratic safeguards that can be misused by the opposition can easily backfire and/or will be seen by more hardline candidates as weakness. The population at that point primarily is driven by more immediate needs than to secure a long lasting democracy with some principles that apply on a meta level when they want food on the table first. Will there be some leaders that want to achieve both? Sure, but they will be careful to pick a particular country to mimic for the practical and political reasons mentioned above.