This answer generally focuses on how to reduce wear on cycling clothing during washing. I am not convinced that spandex or other synthetic fibers would be damaged by chain lubricant or sunscreen (as in I don't think those substances cause the chemical bonds to deteriorate, whereas chlorine will attack some synthetic fibers and weaken them over time; naturally chain lube will stain a lot of things including lycra). I believe the OP's clothes simply wore out during the regular process of washing.
Researchers such as these ones have shown that when you wash clothes with synthetic material, the process washes out little bits of synthetic fiber. This happens every wash. Tangentially, there's a public policy issue here, because those little bits of plastic fiber end up in the water supply, and from there they end up in marine life. That plastic then functions as an endocrine disruptor, with potential consequences for the life cycle of marine life - and we get a lot of our protein from marine life.
Back to the topic at hand. Logically, if the fibers in your synthetic garments are getting thinner and thinner each wash, then the garment will wear thin eventually. Presumably, the thinnest parts of the fabric would go first. Alternatively, perhaps the garment sustained some abrasion in the wash as it rubbed against other clothes, but the general wear process described above could still be a contributing factor.
How might one maximize the life of synthetic garments? It seems like careful hand washing and line drying might maximize garment life. However, this really does not seem practical on a large scale. I just tried it for a couple of my pricier kits. I submerged the clothes with detergent, agitated them occasionally but otherwise let them sit for about half an hour, then I rinsed and line dried them. They smell fine now, so clearly the process worked. However, this did take some time, and the garments did drip a fair amount of water onto the floor. I could have wrung the garments more thoroughly, but I would assume the process of wringing is what creates this microscopic damage to the fibers. In any case, most households are unlikely to have the time or inclination to do extensive hand washing.
One paper found that front-load washing machines emitted far less fiber than top-load ones. Front-load machines are more energy- and water-efficient as well. A webpage for surfers proposes that we also use cool water, use the delicate cycle or otherwise shorten the cycle time, and that we skip the spin cycle if possible. I believe it's well-recognized that you shouldn't run athletic clothes through the dryer, as heat can damage lycra, but regular dryer use should accelerate the general wear process I described above.
NB: line drying as a general practice takes a bit of time out of your day, but it does save energy; I line dry for the most part and use the dryer to selectively freshen up clothes that need it. Using cool water for most or all loads also saves energy, and most new washers and detergents are designed to not require hot water. Also, switching to cool water doesn't require any additional time on your part. So, perhaps readers should do at least this step anyway. You should be line drying athletic clothes, but perhaps readers should consider line drying as a more general practice.
Last, it may also be worth using a mesh bag for delicate items like athletic gear. These bags protect their contents from abrasion by other items in the laundry. This might not affect the wear process I described earlier, but it could reduce the overall amount of wear to the protected items.