Not trying to be condescending at all. Just stating fact. iTunes was the only way to perform updates. Only fairly recently did Apple introduce "over-the-air" updates. However, with the added convenience came the potential that they wouldn't work ... thus, Apple still provides the alternate "old school" iTunes method. Regardless, I absolutely agree with you that the "over-the-air" method is easier ... when it works.
One option is to try doing the "over-the-air" method at another location, not using the same Wi-Fi that you are using now ... like at a coffee shop, or (again) a friend's or relative's house, using their Wi-Fi instead.