I have tried using a Sharpie on tires. For me, it was not a satisfactory solution.
I found the tip too stiff to reach into the recesses of the tire tread, or to reach onto the tire facings within open wheel wells. The result was a tire that was painted the base color of the vehicle, with a black circle on the sidewalls.

Not quite what I was looking for.
I have also tried Sharpies for painting glass surfaces, again without success. Again I found the tips too stiff. I tried to push the tip into the edges and corners without success, and so wound up with glass that was not inked to the edge, but with the rims/supports very heavily inked.
I think the key to success in using Sharpies is to use them for surfaces which are either flat, or raised. Never recessed. So the struts of an aircraft cockpit, but not the glass. Or as part of a camo pattern or national insignia.
Some examples:

The outline of the camo patterns on my French halftracks are done with a black Sharpie. But the wheels and tracks have been painted with a brush, and then washed.
BTW the muzzle of the 105 was also touched with the black Sharpie. I really like the effect this gives, and it is VERY simple.

The Regia Aeronautica national insignia on these (Brand-X) Macchi's were done with a circle of white paint, and then some work with a black Sharpie. Also the radial engines were done with silver metallic paint, and then highlighted with the black Sharpie.
My only suggestion to simplify the painting of tires, is this.
Rather than painting the black around the rim, try first just painting the whole tire and wheel black. Fast and easy. Then, when it has thoroughly dried, come back and paint the wheel within the tire with the base-color of the vehicle. Maybe lighten it up a touch with some white, as you will be painting over black.
I have always found it easier to paint round balls, than circles. So when painting anything that turns into concentric circles, whether it is tires/wheels, or national insignias, start with the outer most ring, paint it as a ball, and work your way in painting balls as you go.