“The public schools in America have no problem with this issues because they use substitute teachers.”
There is one significant difference between the public school system and yeshivas. The yeshivas and their teachers operate according to halacha. This means the yeshiva consults their Rav as does the teacher who may wish to leave during the school year. The public schools operate lihavdil according to local, state and federal laws and the binding contracts signed between the teachers union and school districts as well as the contracts between the members and their unions.