The whole process of electric welding involves forming a plasma arc between the welder and the work.
Once that plasma arc is established, the current path through the arc has a negative resistance, and the current must be limited in some way or the arc will continue to grow to infinite proportions.
Once the arc is established, only a very few volts are needed to sustain the arc, typically around 15v ac or dc.
But getting the arc started in the first place is an entirely different thing.
A typical automotive spark plug with a gap of perhaps only 1mm might need 20Kv to break down the insulating air.
Very fast rise times, and sharp pointed or hot surface reduce the required breakdown voltage considerably. Some welding and plasma cutting torches use a fairly high power RF generator to ionise the air through corona discharge.
This is known as pilot arc.
Sometimes, as with stick welding, you just short the rod to the job, then break the connection, and the inductive back EMF begins the arc process.
While high frequencies, fast rise times, and skin effect may be important during forming the initial arc, although the current will be minimal. It has nothing to do with maintaining the arc, or the actual welding process.
In answer to your question, we don't need the current to flow on the surface.
But using a high frequency pilot arc will cause surface current flow just because it uses a high frequency.
Its just a by product of using high frequency. Its totally irrelevant really, its neither an advantage or disadvantage..