My first goal is achieved! After two and a half weeks I got my passport back from the Kyrgyz consulate in Frankfurt.
The next visa in my queue is the Uzbek visa. My research led to the information that the Uzbek consulate does not require a letter of invitation for Swiss citizens. Thus the requirements for the visa are the same as for the Kyrgyz visa: completed application form with photo, passport, return envelope and that’s it. Let’s wait and see how things will work out.