Can you start birth control pills anytime during your cycle, as long as you take them consistently?

No. Oral contraceptives need to be taken within the first week of your menstrual flow to guarantee their effectiveness. Ovulation can and does occur thereafter, and starting a pill after no previous contraception later than 10 days is like "locking the barn door after the horse is gone".
Yes and No. For cycle control only and not for "birth control", you can start the pills anytime you like and with daily taking of the pill, the cycles will usually correct themselves. However, if you are taking the pills to avoid a pregnancy, you need to be consistent in taking the pills and need to start the first pack pretty close to the first day of your menses.