High or Standard German is not spoken in Kindergarten or the first years of Elementary School. Customarily, Elementary teachers speak to and teach children in Swiss German . This changes at High School level where only Standard German is spoken.
This inconsistency in language use with regards to the education system, and a tenacity to hold on to Swiss German by their instructors even at the start of children's formative academic years, leads to many students not having a mastery of the Standard/High German by the time they reach High School. Hence the extra lessons in order to reach an acceptable entry level of German.
With regards to your comments about English students, I am assuming you are referring to mother tongue English speakers. Naturally, there are many students regardless of what their mother tongues are that need a little help with grammar in their own languages, but we are not talking about the same thing here.
I am aware of a large group of students who study their academic subjects bilingually in English and Standard/High German. Their exposure to German from the onset was that of Standard/High German by their instructors, and when they played at recess, the German spoken among themselves and with their teachers was also Standard/High German (something not normally practiced in Swiss German Schools where everyone generally reverts to Swiss German outside of the classroom). By the time these students were ready to enter High School, the majority of them had mastered the Standard German sufficiently to pass the entrance exams without need of extra lessons. Of course, there were those that needed help in both English and German grammar to pass, but this was not the average problem.
I understand and can appreciate the pride in speaking and maintaining Swiss German as part of an identity, but as it is neither a Swiss official language, nor an official formal and literary language on its own, nor the German that is learned in the Italian, French or Romantsch speaking areas of Switzerland, then in my opinion, it should not stubbornly continue to be used academically, especially at entry level as it may ultimately only be a setback in the long run for further educational opportunities. If Standard German is what is used in texts, reading and writing skills, if it is used as a Swiss national language, then why use Swiss German at all in schools to begin with?