My bike tries to start but won't, and I see white smoke coming from the exhaust
The white smoke indicates that water is getting in the engine, which means water could be in your fuel tank. The reserve fuel comes from the same tank as the main, but the reserve comes from a lower level. The main setting takes fuel from about 2 inches above the bottom of the tank; reserve takes fuel from the very bottom. Water is heavier than gasoline. If you have 1/2 inch of water in the bottom of your fuel tank, it will never get into the carbs if you never use reserve. When you switch to the reserve setting, the water will immediately flow into the carbs.
One way to deal with this is to not add fuel to the tank until after it gets to reserve. This way the water is removed from the tank regularly. The down side to this method is that it ends up in the carbs.
Another way is to use reserve often. This will let a little bit of water into your carbs, instead of a lot all at once. Just remember to switch back to the main position, because if you run out of gas when it's on reserve, you're really out of gas.
Another method is to drain the fuel tank. This method requires more work. You should do this when you buy a new-to-you bike. However, even with bikes that are resurrection specials, and have sat out in the weather for years, all the tanks usually need is a simple drain and refill. Sludge is rarely a problem, but draining the bad gas and/or water is a good idea.
The water problem occurs more often in bikes that sit in the rain. (Strange, but true).