For international students
An internship needs no sponsorship. Stop screening yourself out.
People merge two different questions: can this company hire me, and will they sponsor me later. For a summer internship on CPT the answer to the first is yes, with no sponsorship involved at all. So a posting that says "must be authorized to work without sponsorship" is telling you about the full-time offer, not the internship. Ask a recruiter the precise question: do you hire F-1 students on CPT for summer internships. That gets a straight answer, and it opens doors people close on themselves.