Went thru this same scenario several yrs. back, with questions here as to probable cause.
Tire change and brand change did NOT resolve issue.....same,same.
Added fork brace.......helped reduce the wobble, but was still there.
Replaced the stock roller-ball tye bearings with tapered roller type in steering neck........issue TOTALLY resolved......no matter what tire or how worn it became.
They ARE a bit fiddley getting pre-load drag correct, letting tripple-tree fall off the either side(wheel mounted) on its own with a slight drag on it once 'pal' nut is tighted.
I know in the track world, riders like the ball type for the supposedly 'road feel', But for the life of me don't know why Honda chose ball type in a location that absorbs road shocks constantly, mostly while in straight-ahead positioning
. Even the Wings retain the ball type.
On a side note, my 33yr.old XS11 Venturer with 82K original miles still has its original tapered roller bearings in steering neck with no 'notchy' in center position, and bike is only 35lbs. in weight lighter than the ST11 with same 82K on it(62K when changed to tapered roller). Some of that longgivity difference may be attributed to the fact that the dealer(still in business) I puchased the Yammy from was anal particular when uncrating a new bike. They would remove tripple tree and CORRECTLY pack bearings with grease since all the Jap. manufacturers had minimal lube in that location(and still do today) prior to shipping.
Just was something I became aware of decades ago on moto-x bikes, and definitely was the norm on street bikes
.