Horizontal dropouts are what allowed me to calm the steering response of my 1979 Fuji Professional.
It's a big 59cm and with steep angles, though fits me perfectly with the original, oddly-short (9cm) Nitto Crystem (but which leaves relatively little weight on the front tire).
Moving the axle back 1cm (after photo was taken) definitely made a quite-noticeable improvement, now it's a favorite ride of mine!

Horizontal dropouts also can accommodate wheel-clearance issues, such as when a frame gets bent or a spoke breaks out on the road!
Also useful for accommodating fenders in some cases.
One concern might be that the rear brake pads may contact the tire after axle is moved, so re-positioning the pads may be necessary.

One more (theoretical) advantage might be that a derailer mounting claw can be used in a horizontal dropout slot if perhaps the original hanger becomes damaged or was "Drewed".