Turns out Minerva has this "fun" LESS rule in its skinStyles/mediawiki.special.userlogin.common.styles/minerva.less file:
// Hide keep me logged in checkbox on mobile. It will be marked as checked by default. #userloginForm { .mw-htmlform-field-HTMLCheckField { display: none; } }
This works fine as long as extensions don't add any checkboxes, and if they do, at least they don't add any mandatory checkboxes. Coincidentally, the NewSignupPage extension does just that: it adds a mandatory checkbox that users must manually check in order to be allowed to register. Since the LESS code hides the extension-created checkbox, it essentially breaks mobile signups completely, forcing users to use the desktop version in order to sign up.
Reported on ShoutWiki as osTicket #3583.