The first thing that needs to be said in answering this question is that there is a big difference between being a “stern disciplinarian” and accepting what is going on at this sandwich shop. Requiring workers to come to work on time is not the same thing as being a stern disciplinarian. Coming on time is a minimum requirement for holding a job. Therefore, you can and should demand that workers come on time and you can do so without being a “stern disciplinarian.”
The way I would assert myself as boss is by calling a meeting of all the staff. I would say something like this. “Since I’ve been hired as manager here, I have been noticing a few things that need to change. The main thing is that we have been having too many people coming to work late. Being on time to work is something that is expected of every worker at every job in this country. You all know that, I’m sure. So, I just wanted to let everyone know that we will not tolerate people coming to work late. I’ll be here when I am scheduled to be here and I look forward to seeing each of you when you are scheduled to work. “
If you say this, you are essentially asserting your authority in a subtle way. You are not challenging anyone or saying “you have to respect me.” You are just stating a simple fact about what is expected of the workers. You are demanding respect simply be talking as if you expect to be obeyed. This is a much more effective approach than blustering or talking about how you have the authority to fire them or anything like that. In this way, you can set reasonable expectations and you can do so in a way that is not “in their face” or offensive. This will assert your authority without requiring you to be a stern disciplinarian.