Is it your intention to have the Mystic’s +4 to find secret doors apply to the 18+ to find them casually?

Can’t answer for the autarchs but my guess reading draft version 10 is: a mystic thus finds secret doors at 14+ if he is actively looking for them or at 18+ if exploring an area without specifically looking for them (simiar to an elf - CR p.92)

Beastman has it exactly correct. If actively searching for secret doors, he gets +4, reducing his target value to 14+. He can casually notice secret doors on an 18+ (normal folks can't notice them at all without a detailed search).

This is effectively the same as the old B/X rule for an elf: He notices secret doors on a 6 casually and on a 5-6 if searching.