Am I setting my sights too high?
I'd say no, you can set your sights as high as you want.
Is it unreasonable to expect to find a senior position without the leadership requirement?
I would say that all Senior positions involve at least a bit of leadership.
Maybe you say you haven't had "official" experience on that, but I am positive that the process of getting to Senior position on your last job taught you something about leading, organizing, delegating, etc.
If not a Project Manager, I am sure you have some degree of experience in leading, so I wouldn't consider yourself as having zero knowledge on that. Just remember to keep truthful and refrain from boasting or exaggerating your real skills to avoid compromising your application.
Am I better off applying for mid-level/standard positions and if so, how do I explain the inevitable interview question "You were a senior, why are you applying for a mid-level position?"
If you did this, the answer to that interview question would be "I am applying to a mid position because I don't feel qualified for Senior ones, even though that was my last job"... perhaps you are overthinking this one.
To be honest, we may be in a job title relativity situation here: job titles vary greatly between companies, as the concept and responsibilities of Senior Engineer can vary much or less depending on the context and specific company.
Bottom line, I say you don't close that door to yourself yet. You have been Senior Engineer before, so you could proceed with the recruiting process and decide if you really are fit for the role once you know more about what it really involves. Only then, you feel that the managerial/leadership is too much for you consider declining.