I once had a piece of road granite stuck between brake pad and disc and that made an awful noise, but the noise was there all the time - don't remember if I had to prise it out or got rid of it by reversing and applying the brakes. If the noise is a 'shriek' and only there when you apply brakes it is 99% going to be the noise of the acoustic wear strips on the brake pads,
you have to inspect both pads on every wheel as the wear on the outer pad that you can see may not be the same as the inner pad. The hydraulic cylinder only pushes on the inner brake pad, brake caliper floats on two guide pins ( to allow it to equalise the pressure on both pads ) that are supposed to be lubricated and shielded from dirt and water by rubber bellow seals, sometimes the pins are just not lubricated, and sometimes the protective bellow have slipped of or been damaged ( split ) and allowed water and crud in. If the caliper is seized on the pins the inner pad will wear a lot more than the outer one because it is the only one touching the disc, and the other pad will hardly wear at all.
Another explanation is that the anti-squeal shims behind the pad have been left out, this will make a different noise ( remember the noise buses and big lorries used to make when braking at low speed, that was the pads vibrating). That used to have a lower pitched tone a bit like the foghorn you describe...
https://auto.howstuffworks.com/auto-parts/brakes/brake-parts/brake-shims.htm