From teacher to teacher
The intention of the Choosing the correct case pages is to give beginning German learners a clear decision-making process that they can apply when doing written work. When speaking German, of course, there is no time to apply the explicit cognitive processes recommended here. Students need to be reassured that getting case grammar correct in spoken German is a demanding task that requires significant exposure to both oral and written German, as well as the acquisition of a large vocabulary that includes knowledge of noun gender.
The complexity of producing accurate German in respect of the cases should, in my opinion, be reduced for beginning learners of German. It is not surprising that they make wild guesses at the correct form if they have to conjure it up from all the following variables: case, article or determiner type, number of noun, gender of noun.
Initially at least, I believe that they should be given three of the four variables and told to produce the fourth from memory. Typically, this would mean giving them the appropriate case table and the gender of the noun. They would then only need to determine the correct case for the given situation. Alternatively, they would be given the case and gender of the noun, and need to determine the correct form of the article (and adjective).