Anisotropic, I took the liberty of exploring the UCLA website a bit, and I think it would make a lot of sense for you to contact their first-year offices (call in for a conversation with a real person, not just their email contact form) and have a conversation. I saw evidence that they would be willing to consider extenuating circumstances with regard to the courses you did not finish this past school year, and that they can work with you on solutions.
Also, they appear to be planning on mainly online and hybrid classes for the fall, which means there is a good chance you could have a somewhat more flexible college environment.
And as to some of your perceptions about where you rank intellectually, I would point you to the research on imposter syndrome, with findings that a surprisingly high number of high achieving persons believe (inaccurately) that they are not as qualified for their roles as their peers are, and that "if people only knew," they would be exposed as frauds. Here's a readable and practical lay article on this published recently in Science (AAAS), and written with a target audience of practicing scientists:
https://www.sciencemag.org/careers/2020/07/how-banish-impostor-syndromeIn the small sample of your accomplishments that you have referenced in your communications with us, it is apparent that there are objective, external data that support your high mathematical ability, which is why I maintain that this should not be a concern. It may be, though, that some of the markers you rely on for external validation of your ability are weaker in sensitivity or specificity, or have confounds (e.g., admission to a specific post-secondary institution), but have outsize impact on your perceptions. These are conversations that it might benefit you to have with your therapist, regarding interal and external sources of validation, relative weighting of different data, and developing resilience for unexpected outcomes.
You have a tremendous amount of potential (based on what you have already accomplished, especially in the face of obstacles), and many possible paths to a fulfilling life of intellectual and relational satisfaction. Whether you enroll in UCLA this fall, or take a gap year, I would encourage you to continue challenging yourself not only intellectually, but also in your understanding of yourself and others.
Best.
