1) They'll scratch up: first off scratches can reduce light transmission but solar panels don't require good "optical quality", only transmission; the light is free to scatter on its way in. It's the same thing that applies to greenhouses - you may have noticed that many greenhouses use "fogged" plastic that you can't see through, yet still lets the vast majority of the light in (in that case, the scattering is actually seen as advantageous). Beyond that, in the case of roadways, I'd think it a given that they'd coat them with a an anti-scratch coat (aka harder than Mohs 7 / quartz sand, the hardest common natural material))
2) Traction: Traction glass exists - it's just surface texturing. They use it for semi-transparent flooring, it's nothing special.
A thin flat clean surface is the most efficient cover for the cells. Any deviation will decrease the efficiency. You are suggesting a rough thick 'milky' material with scratches on it. It will scatter a lot of the light away from the cells. Greenhouses are not a good counterexample as they are not built for *maximum* throughput, just for one that delivers a stable 90F atmosphere inside.
3) "Glass would break and then shred tires": It's easy to make glass bear purely compressive loads (solid objects on both sides of it) without fracture - that's what it's best at. It's shear and tensile loads that glass is bad at, but these aren't applicable when it's flat on a hard surface. And lamination, like in windshields, prevents dangerous shards from coming off in the event of a fracture. This is not an actual limitation.
But the glass will not bear purely compressive loads. There will be impact forces of heavy objects falling on it at high speed, cars driving over hard pointy objects lying on the road (stones) and ice expanding within the grooves between the tiles and underneath them (this is the greatest nemesis of the asphalt road).
3) Shadowing: Go to Google Maps satellite view and look up random roads. The overwhelming majority of road surface is completely unshadowed at any point in time. Even in-city roads are overwhelmingly unshadowed. Shadows are practically irrelevant in the countryside except in wooded areas.
4) Costs: The costs of the materials for a road are a minority of the costs of the project, and continue to be a minority of the cost of the project under any realistic pricing for large-scale production of paving panels. A key driver for affordability, however, would be scale: this means large scale production (so road panels are similarly priced to rooftop panels plus the extra glass costs) and continuous paving systems. Anything smaller scale would have elevated costs.
Two problems - complexity and maintenance. A solar road is orders of magnitude more complex than a regular road - first it will drive up the cost because it's not as simple as pressing a malleable material onto a rocky surface. You will need to connect the panels and lead wiring, construct maintenance access points, test the functionality. Maintenance will be a major pain in the ass. And for all of this you will have to hire more expensive technicians than what you need for regular roads. Scale does not help too much either. Regular solar panels are already mass produced and are orders of magnitude more expensive than asphalt. Even if you cut the price in half somehow, it will remain orders of magnitude higher than regular road surface.
5) "They'd be better on roofs": the main problem with roof installations is there is no way to do mass-scale continuous install (the sort of possibility that paving gives). Each roof has to be handled on its own, with its own engineering issues, with its own project overhead, its own inverters, etc. The key issue to cost reduction these days is getting rid of the overhead; panel production costs themselves have gotten quite low and keep going down. Furthermore, with a road you get "two birds with one stone" - a driving surface and a power generation surface built at the same time in the same space, sharing the same project overhead. It's fine to sacrifice some panel efficiency to glass, shadows, dirt, etc if it reduces your overhead costs.
Only if you put them on separate roofs. Constructing a roof over the road would give you the possibility to install continuously on a mass scale. You coud also tilt them at the most advantageous angle. It would take more effort to raise the pillars but if what you say about the 1/3 efficiency s true, it would TRIPLE the power output at possibly the same price (you'd have to add asphalt and pillars, but you could use regular cells that are not expected to bear any significant load).
Number one on my list is the snow-melting concept. It takes five minutes to run the numbers on that and find that it takes way more energy than could ever be considered reasonable. You could melt thin layers of frost off the surface, but nothing of any relevant mass.
If one wants to pursue an anti-snow approach, my personal alternative is having an air blower in your (already required) regularly-spaced inversion substations, blowing air into the wiring conduit, with small regularly-spaced holes in the panels. You'd get a weak "air hockey table" effect over them, potentially enough to help divert snow off the shoulder (although far too weak of an effect to have any effect on cars). The energy calculations for that show that it's actually plausible - and air blowing through the panels (cooling) would increase their efficiency a bit, possibly even paying for the blower energy consumption.
Adding moving parts into an already complex outdoor system is just asking for more trouble. You could not just put exposed propellers like the windfarms have on the side of the road as these would pose a safety threat. They'd be quite expensive to construct and maintain. Plus, if you do the math, a typical office fan uses 100 Watts of power. You'd probably need something stronger than that - maybe 300 Watts. Every three feet or so - so 100W per foot (and even that is pretty optimistic). A typical solar panel gives you 10W per sq foot, in your case, 3W/foot. With a standard 24 foot road, you'd still be short and you'd have to supply the system.
Again, a roof-over-the-road system would not have these problems. Actually, it would decrease the snow removal problem!
An opposite snow-removal approach would be to make use of the glass to trap heat in the ground on sunny days, which can then radiate out and cause snow to melt faster. The obvious downside is that it'd be hard to develop a system that doesn't involve the cells also running hot, and thus less efficiently.
Yeah, the other problem is that there might not be enough heat for that for a couple of months regardles of how much of it you trap.
If one wants to go more extreme on their cell cooling, another possibility arises. One could combine the need for surface texturing, the need for the glass to begin with, the cooling, etc and have the surface of the road be a fresnel lens concentrator, focusing light on higher efficiency, more expensive, but smaller cells as the receivers. Solar cells - if properly cooled - operate more efficiently at higher concentration levels, and with higher efficiency cells, your road could net significantly more power production. But it does mean more complicated panels.
This if fundamentally wrong. You have a certain amount of light falling on the surface of the road. Concentrating that light into smaller surfaces (with the lenses) will not produce more heat. You'd have to construt a lense that's larger than the road to collect more light onto the same area.
Meh, there's a solar bike path [google.is] in the Netherlands and they don't seem to have excessive problems with dirt. Because rain exists. They got significantly higher generation than they were expecting - only about 1/3rd less than what you'd expect from rooftop mounted panels.
A bike path is a nice idea, but it's far from the robustness a road would require. And 1/3rd is actually quite bad. It's less than what the first solar panels from the 50's were collecting.