Short term: maximal flexibility and power
As an example let's imagine a software tool that does something. What exactly it does is not important, but let's assume that it needs to be configurable and customizable. A classical approach to this is to create a small core and then add a configuration language for end users. The language may be Turing complete either by design or it might grow into Turing completeness accidentally when people add new functionality as needed. Most commonly this happens via some macro expansion functionality added to "reduce boilerplate".
Perhaps the best known example of this kind of tool is Make. At its core it is only a DAG solver and you can have arbitrary functionality just by writing shell script fragments direcly inside your Makefile. The underlying idea is to give developers the tools they need to solve their own special problems. These kinds of accidental programming environments are surprisingly common. I was told that a certain Large Software Company that provides services over the web did a survey on how many self created Turing complete configuration languages they have within their company. I never managed to get an actual number out of them, which should tell you that the real answer is probably "way too many".
Still, Make has stood the test of time as have many other projects using this approach. Like everything in life this architecture also has its downsides. The main one can be split in two parts, the first of which is this:
Even though this approach is easy to get started and you can make it do anything (that is what Turing completness does after all), it fairly quickly hits a performance ceiling. Changing the configuration and adding new features becomes ever more tedious as the size of the project grows. This is known as the Turing tarpit, where everything is possible but nothing is easy.
In practice this seems to happen quite often. Some would say every time. To understand why, we need to remember Jussi's law of programmers:
The problem with programmers is that if you give them the chance, they will start programming.
An alternative formulation would be that if you give programmers the possibility to solve their own problems, they are going to solve the hell out of their own problems. These sorts of architectures play the strings of the classical Unix hacker's soul. There are no stupid limitations, instead the programmer is in charge and can fine tune the solution to match the exact specifics and details of their unique snowflake project. The fact that the environment was not originally designed for general programming is not a deterrent, but a badge of merit. Making software do things it was not originally designed to do is what the original hacker ethic was all about, after all. This means that every project ends up solving the same common problems in a different way. This is a violation of the DRY principle over multiple projects. Even though every individual project only solves any problem once, globally they get solved dozens or even hundreds of times. The worst part is that none of these solutions work together. Taking to arbitrary Make projects and trying to build one as a subproject of another will not work and will only lead to tears.
Long term: maximal cooperation
An alternative approach is to create some sort of a configuration interface which is specific to the problem at hand but prevents end users from doing arbitrary operations. It looks like this.
The main thing to note is that at the beginning this approach is objectively worse. If you start using one of these at the beginning it is entirely possible that it can't solve your problem, and you can't make it work on your own, you need to change the system (typically by writing and sending pull requests). This brings enormous pressure to use an existing tweakable system because even though it might be "worse" at least it works.
However once this approach gets out of its Death Valley and has support for the majority of things users need, then networks effects kick in and everything changes. It turns out that perceived snowflakeness is very different from actual snowflakeness. Most problems are in fact pretty much the same for most projects and the solutions for those are mostly boring. Once there is a ready made solution, people are quite happy to use that, meaning that any experience you have in one project is immediately transferable to all other projects using the same tool. Additionally as general programming has been prevented, you can be fairly sure  that no cowboy coder has gone and reimplemented basic functionality just because they know better or, more probably, have not read the documentation and realized that the functionality they want already exists.
 Not 100% sure, of course, because cowboy coders can be surprisingly persistant.