Would the Wright Flyer have flown better with thicker airfoils?
There were several other limitations on the first Flyer besides the airfoil -- the engine weighed four times what the brothers had been promised it would (because the block was cast, rather than fabricated from sheet), and would have been underpowered by today's standards even if it had delivered to spec.  The whole machine came in just under 300 kg; a modern ultralight that size would weigh less than half as much and have nearly twice the power.

Given those limitations, the airfoil is a relatively minor consideration.  Yes, there are modern airfoils that would perform better in terms of coefficient and lift-to-drag -- but the thing that would have helped the most would have been saving 40 or so kg. on the engine weight.

It's worth noting that through the First World War, with aircraft flying at four or five times the speed of the first Flyer, at altitudes up to 4000+ m, inverted and other aerobatic maneuvers, what we'd call relative thin, cambered airfoils persisted.  Look at the wing section of a Sopwith Camel -- though slightly thicker and more deeply cambered than the original Flyer's airfoil, it was still thinner than anything you'll find today except on single-surface hang gliders and ultralights.

How thickness and camber affect flight characteristics is subject matter for a full length college level textbook, but short version: more camber tends to increase lift coefficient over some range of angle of attack, but increases drag in proportion; adding thickness increased coefficient over a broader AoA range, with less drag increase, but has less effect at low AoA than increasing camber.  There's a compromise, with both increased by similar amounts, that includes the well-known Clark Y, NACA 6409 and 6412, and many other common airfoils used in general aviation, especially when wood and fabric were the building materials.