AFAIK, private is the default everywhere in C# (meaning that if I don’t write public, protected, internal, etc. it will be private by default). (please correct me if wrong).
This is not true. Types defined within a namespace (classes, structs, interfaces, etc) will be internal by default. Also, members within different types have different default accessibilities (such as public for interface members). For details, see Accessibility Levels on MSDN.
Also,
So, what’s the reason to write that keyword, or why does it even exist?
Specifying this explicitly helps denote your intention to make the type private, very explicitly. This helps with maintainability of your code over time. This can help with other developers (or yourself) knowing whether a member is private by default or on purpose, etc.