OMG! Most of the population hasn't the tools for thIs complicated and detailed work. Most of the population don't even know how to remove the headlight assembly.
How about someone here offer a product that we can buy to apply with a rag and then wipe it off. What product removes at least some of the haze and only takes minutes for the job - not hours and lots of tools?
Many people may tell you a specific product that works well, or a little faster, but often it's just splitting hairs. If you want a simple, time tested way to restore and maintain them it is not so different than refinishing and protecting anything else.
You consider the material density and amount that needs removed, and select the appropriate abrasive. For many that will just be a fine grit of wet/dry sandpaper, then switching to a finer still grade, used wet so it doesn't clog up and because this is plastic, you want to keep heat down, so if you are going to use motorized tools, don't stay in one spot for too long but this is entirely do-able by hand with no tools. Do not apply a lot of pressure with the lower grit # sandpaper, expecially not in hot weather, to avoid getting abrasive particles embedded in the plastic. More passes instead of more pressure. Mask (tape) around the headlights to preserve the surrounding areas.
After the finest sandpaper, use a polish. It could be plastic polish. It could be metal polish, if fine enough grit to be designed to truly polish. It could even be toothpaste if you want to get more exercise polishing longer or are using a power tool. A toothbrush can also come in handy if your lens has markings or other details molded into it. Final step is apply a polishing wax or sealant type product, same as designed for paint. Any cheap product will work and as on paint, some will last longer than others with sealant lasting longer than wax.
Once you have done this, the first and only time after waiting too long, from then on you can keep up with the rate of decay, can quickly and
easily touch up the surface by simply hand-applying another coat of polishing wax or sealant after a car wash.
So there you have it, it only takes minutes to hand finish and seal your headlights with inexpensive materials found at any hardware and automotive store, and these same materials are versatile for many other uses too, a pack of good (I prefer 3M) sandpaper and polish would last decades for several vehicles if you don't have any other use for it, though some sealants will shelf rot before then.
A clearcoat paint will last longer than a wax/sealant if you did nothing to maintain them, but then when it degrades it takes longer to refinish again to get rid of the clearcoat paint and reapply, than if you just do as mentioned above, put more polishing wax on every now and then after a car wash instead.