Steering damper has no affect on this problem. It only comes into play during rapid rotation of the steering, not slow oscillations.
Voyager is correct that steering damper does not inhibit left-right turns of wheel during normal turning, but only dampens rapid oscillation of steering (speed wobble/tank slapper.) It sounds like that may be what Ian is experiencing, though--rapid, scary oscillation, and that's exactly what the dampening is designed to prevent. It's easy to check if the dampener works--review the video--and pretty easy to fix if it doesn't work.
Of course, one should always review what, if any, changes have been made to the machine prior to the issue developing. Clearly, if you change the tire(s) and then get a wobble, checking the tire(s) should be first on the list. If you didn't change anything and the issue developed, it's likely due to deterioration or failure of something on the machine. It sounds like it's been riding fine for a while, and only recently manifested this wobble on slowing to a stop.
Reviewing this forum, I see several lengthy discussions of the issue going back to the mid-2000s. Frequently tires, tire wear, tire balancing, tire pressure, are implicated. Also common is the steering damper, and less common are steering rake (2005+ models), rear drive alignment, etc.
If you trust your mechanic, seems like changing the tire is best guess. I'd still check the damper.