In the world of bike racing, strength training is the religious war. When the weather gets cold, coaches, trainers, and athletes head inside and start to beat their keyboards (in place of their chests), sling insults, and proclaim how strength training is the prodigal son or a false saint.I could leave you in suspense as to my position, but I won't. I sit right on the fence.
(I'm sure that was totally unsatisfying if you were expecting a tirade for or against strength training. But the reality is, it all depends. Yes, like most things in life, the issue is not black or white.)
First, the argument against strength training. For those of you who love strength training, you'll hate this part. For the scientists and data junkies, login to PubMed and get your search on. (1) There is a lack of research showing definitively that strength training helps with cycling performance. It's easy to cherry-pick studies that show an improvement in VO2max, increased threshold, maximal power production, etc., but they are limited, have flaws in methodology, don't show significant differences, and/or haven't been reproduced. (2) The lack of specificity in most strength training means it doesn't translate to cycling performance well. (3) It takes energy to do weight training, and that is energy that could be used to do more specific training or to recover faster. (4) Cycling is an aerobic sport. That's it. An athlete's maximal strength has little to do with cycling success. Even at peak wattage, a cyclist is producing only about 50% of his/her peak force. Increasing this wattage further is a product of improvements in the person's energy systems, not muscular strength.
Next, the argument for strength training. (1) If you can't physically turn the pedals, you need to get stronger. And you probably have a hard time getting out of bed and walking to the bathroom. I'm not sure you should be worrying about riding a bike in your condition. (2) You can't generate power if your core isn't strong enough to support/stabilize the force your legs produce. Well, actually, that's not very relevant since generating maximal force is not really required to perform well in a sport whose success is based on your cardiovascular performance. (3) Maximal force is correlated with standing start performance. So you're a kilo rider? Oh, you only ride road races. Did that 50m gap at the start line ever win you a race? (4) If you're injured, you can't train....
Wait, what was that? If you're injured you can't train? Since when!? Since I've been lying here with my herniated vertebral discs. Since I pulled a muscle in my neck. Since I tried lifting a box my wife made me move and I may have torn my biceps. Since I broke my hip after tripping over a toy on the floor.
OK, I've been a little harsh on the strength training. It really doesn't help your cycling. But it can help you maintain muscle balance and be a part of good abdominal and back health. It can be part of a general strength and stretching routine that helps you get through life without injuring yourself in daily activities—or carrying your track equipment between the car and infield. And it can be a good weight bearing activity to help ensure you develop and maintain good bone density (cycling alone is great for cardiovascular health but often leads to decreased bone density if its the sole activity for a person).
So where do I come down on strength training? I think it is a good off-season activity for cyclists because they need the strength and suppleness to get through everyday life so they can keep riding consistently. And for long-term bone health, cyclists need to do a weight bearing activity. But for cycling performance, ride smarter. That may be better form on the bike, more hours, less hours, more intensity, improving 1 minute power, ... whatever. But you'll find the time and energy you spend on the bike will pay off more than the time and energy you spend doing strength work ... just as long as you stay healthy enough to stay on the bike.