I recently graduated with a bachelor's degree and have been looking for part-time research positions since I can't find a job. I have not heard back from any professors at institutions other than those at the ones I attend. Why is that?
All of the previous answers are correct. Plus: Student workers also have a different legal status. One can, for example, hire international students as student workers even though they are not allowed to work because they do not have a work visa. But that is only allowed if they are hired at the university they are enrolled in.
That underlies a general principle: You can only be a student worker at your own university; at other universities, you'd just be a regular employee with all that entails.
In the US context, it's typically impossible to employ someone as a research assistant (a specific job title) unless that person is already a student at that university. Funds for graduate student research assistantships are often specified in grants with the understanding by the university and the funding agency that this money will go to graduate students at that university.