You were right in your understanding that schools, hospitals, and medical centers are locations that you can find a speech language pathologist. You can also find us in jails, cancer centers, the oncology department at a rehab hospital, dental offices, voice centers, neurological research labs, an ENT's office, private practice, etc.
So to answer your first question, what is the typical place, there is not a typical place. You can find us many places :)
That is awesome you want to work with children. There are so many ways in which you can do that. You are not limited to a school as there are children's hospitals and camps but I will focus on your question regarding your chances of getting a job in a school. That depends on your clinical experience and your performance in graduate school. I have no doubt you are a solid student and can represent yourself in an interview.
Schools need speech language pathologists so those jobs are not going anywhere. What you may want to narrow your focus on is where would you like to begin? There are many different types of schools. Are you interested in the autism population? There are entire schools dedicated solely to students on the spectrum. Are you interested in severely handicapped children? children with Down's Syndrome? children with swallowing disorders on feeding tubes and trachs? There are so many specialized schools that allow you to find your concentration. The wonderful caveat here is our field is rich in that you can change course in your career and still stay within the field. It is impossible to get bored with the amount of ground we cover. There are so many children than need your help! Thank you for your interest in the field.