Should a teacher be able to solve all the assignments they give their students themselves?
In general, yes, a teacher should know how to do any assignment and, in some cases, should have actually done it. In teaching programming, for example, it is usually a mistake to assign a problem that the instructor hasn't essentially done themselves.
The reason is that one of the tasks in making assignments is to estimate the effort and time required to do the task. If you can't solve the assignment yourself, you are giving an open ended task to students. In almost all situations students have a limited time to spend on any given assignment and that competes with their other tasks.
There is an exception to the above. If you tell the students at the beginning that you are assigning something you don't have an answer to, and will be grading their efforts rather than their results, then you can make this work. The assignment becomes an exploration. At lower levels of education this makes less sense than it does at higher (say, doctoral) where it is natural to explore the unknown.