If you don't like begging for more money, you could do a couple things:
* You could find a lord willing to give you a bigger tax income.
* You could just recruit whatever troops you can afford, without begging for more money.
Constantly asking for more money is NOT a requirement for playing the game. There is nothing in the game that requires you to constantly be asking for more money. If you choose to do so, then that's your choice.
I agree with you on all that, but the title of the thread is Improved Army Chest, so I am going to ignore talking about that since I agree and give other suggestions.
What I would like to see are options. To elaborate on it, I mean options for what the sponsor pays. A poor sponsor may only provide gold for training, while richer sponsors will pay for equipment damage as well and the richest will even pay sea travel, instead of all of it automatically coming from the army chest.