The Labor and Welfare Administration (NAV) of the Norwegian government is doing exactly that, we have been opening up our code since 2018 and are accepting pull requests across our 2000 public repositories from those who want to improve government services https://github.com/navikt this has also enabled a much more close collaboration between new and existing partners that was previously unheard of!
I think the main reason why this is not done actively in Germany, is missing people with expertise. Lots of old guys from the pre-internet era taking care of the technical systems. Unfortunately, the gov is not doing much to make at least official positions attractive to young programmers. And of course, if it's done by a contractor, they do not wat to share any code with the public.
This might be because the government is a really unattractive employer both in terms of culture and remuneration.
However I have met some truly brilliant people working for the city of Berlin. They're just limited by red tape and diffusion of responsibilities, not to mention the amount of work that needs doing.
Opening the code up to pull requests could give the city IT a free boost from motivated citizens, I think.
