Well, the buttons are disappearing because the elements are being assigned a background color and a foreground color, but the links don’t have any text, so the background of the link ends up on top of the text, which is positioned behind the link.
This exposes a considerably *bigger* accessibility problem on the site – that none of your buttons have accessible link text. The text that’s shown is contained in a <div> element that’s positioned behind the link element.
As a result, a user with a screen reader trying to navigate through those buttons would be unable to identify the purpose of any of the buttons. Switching to high-contrast mode just makes the accessibility of the site the same for sighted users as it already is for screen reader users.