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I'd say most mental barriers unique to international students can be summarized as:
"You're right that by the time they've assumed, it's often too late to change their minds. But you're wrong if you think you can't change their minds at all. You just have to be one step ahead of them.
"It's easy to presume a company's staff make all hiring judgment calls on their own just because they are in a position of authority to do so, but that's a myth. They know what expertise they don't have. You will be amazed how open-minded and grateful hiring officials are about being told how to handle your handicap."
"It's hard, and it will take a lot of practice to get good at, but investing the time in getting good at 'making your own handicaps look like strengths' will drastically change your job-hunting life forever. It is the foundation on which everything else you're going to get training and practice for as you job-hunt (networking, cold-calls, resumes, cover letters, interviews) will become more successful."
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I suppose I haven't yet said anything that isn't true for domestic students, too ... but what I think is unique about international-student-specific mental barriers is that they ALL seem to be variations on this theme. So, below are 6 specific complaints of this type that I frequently hear international students worry about and the 6 specific 'So make it sound like a good thing' advice I give them.
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You're right that they're judging you as soon as they see your immigration status - so the best you can do to 'jump ahead' of that is in your cover letter / resume. Some companies legitimately don't want to work with F-1 students, but in my experience, most companies simply don't know the difference. Here's something you can throw in as a footnote to your cover letter / resume if you're explicitly getting shut down by companies with, 'No sponsorships!':
'Over the summers and for [12 / 12-27 (if STEM)] months after graduation, I am authorized to do work in America that gives me real-world experience extending my academic training in [list majors] at no cost to your company. Hiring a current or recently-graduated international student is different than sponsorship (which pertains to long-term work visas) and does not require any complicated legal filings or fees. My work for your company would only be constrained by duration and subject area - hiring me can be structured as an internship, a temporary position, or even a direct hire. Please ask me if you have any questions!'
'I am an international student from [country] who intends to establish a career in my home country. My visa allows me to perform [major]-related work during the course of my studies and up to [one / two and a half (STEM)] year[s] after graduation.'
But you can gain relevant skills at a job even if the job isn't specifically structured around a skill-building goal.
You can also work jobs that are a 'temporary' position or a 'direct hire,' as long as you stay within the duration, subject-matter, and hours/week requirements of your F-1 visa status. So broaden your job search beyond internships.
'Cold-call' relevant staff at companies you'd like to work with asking them if they ever have, or can create, 'short-term work' - rather than asking 'if they have internships' available. You'll find that many of them who 'don't have interships' are open to 'having short-term help.'
Open yourself up to working with temp agencies (just make sure they understand your duration / subject / hours-per-week limits so they don't suggest inappropriate placements).
And here's a nice little phrase you can add to cold-call letters or your resume to ensure they don't make any assumptions about whether or not you're easy to hire:
'I am an international student from [country] looking for hands-on experience in America before embarking upon a career in my home country. My visa allows me to perform [major]-related work during the course of my studies and up to [one / two and a half] year[s] after graduation.'
Create a space on your resume called 'Projects' or 'Projects and Research' or 'Projects and Field Work' - etc.
Include a 1-sentence description of a research paper you did. Describe a class trip by what you did on it, e.g. 'Investigated the pH of lake sediment along Lake Superior boat docks.' Etc.
The #1 thing to remember to be successful at this is to NEVER let the fact that everyone in the class had to do something, or the fact that the professor walked you through how to do something, stop you from mentioning it. Your resume is about advertising YOU, and you really have done these things!
Include even things you couldn't repeat without guidance. You're young. If a company needs you to be able to do such work independently, they'll let you know at the interview stage. But don't be afraid to list something you've done in class just because you felt like it was 'no big deal.'
CONCERN #4:
Different companies put different values on the resume and cover letter, but here's a secret: those who actually read every applicant's cover letter REALLY love cover letters and get a whole lot of generic ones they hate.
If you use your cover letter (and any free-text areas in the application form) to show that you actually bothered to read the job description (use the job description's words!), you will be in the top 50% of cover letters.
If you also follow up the company's own words with descriptions of why you are the candidate who can solve those needs, you will be in the top 5% of cover letters - including Americans applying to the same job.
Most people get too busy applying for every job they see to tell the story of why they personally fit the job in question. Take the time and you will be at the top of the stack, despite your visa status, employment background, lack of experience with American customs, etc.
Develop 'elevator-speech'-length answers to them (15-30 seconds).
Think about how you would explain what these questions are asking to your grandmother or to a 10-year-old. Now bring it down to 15-30 seconds.
Phrase your answer something like this:
'I haven't had the opportunity to work with [______], but I know that it is a [_____] used for [______], and that when working with [_____]s, it is important to make sure that issues of [______, ______, and ______] are addressed because otherwise, [____, ____, and/or _____] could happen.'
