Essentially, the paragraph states that for suicide, those who did not receive gender affirming care saw a 3x higher suicide rate than controls - and this is with overcontrolling for psychological treatment visits. Those who did receive care had no significant difference in suicide rates from controls. Dr. Meyerowitz-Katz, epidemiologist, stated of these findings, “The authors in their discussion focus on the fact that this difference was not statistically significant (presumably the p-value was 0.051-0.054), but that's not a useful distinction. There's a lot of uncertainty here, but the increased risk is still remarkable!”
Notably, this is the only section where the researchers withhold the model that doesn't include visits to psychological specialists. It's likely that the correlation between receiving gender-affirming care and a decreased suicide risk would be even more pronounced in a model free from the issue of overcontrolling. If the researchers had presented such a finding, it would fundamentally challenge the basis of their paper... that gender-affirming care indeed saves lives. Even in attempts to dilute this relationship with confounding variables, the signal around gender affirming care remains strong!
Main problem with the student explained in that quote. The overcorrection problem refers to the fact that suicide correlates with psychiatric visits because suicidal people are more likely to seek help and therefore would be like saying visit to cancer doctors causes death from cancer.
Additionally, the median age for referral to the gender clinic was 19, so the results can't really be applied to minors.