Why doesn't Vanguard allow to pay fees pertaining to 401(k) accounts using resources located outside the 401(k) accounts?
Lets assume for a moment that the law would allow external funds being used to pay account fees.
The 401K company would see it as an added expense, and not want what you propose.
Shortly after each payday for your company the 401K service provider receives an electronic transfer of a sum of money and a data file explaining how the money is to be split among the employees. The service provider's computer then allocates the money to each investment choice previously designated by each employee.
Now imagine that they need to bill each employee and accept payment. Each employee could send a check. If each account was charged a flat fee that would still result in an extra transaction per customer. It would be even harder if the fee was based on the size of their account. Each bill would have to be generated, and sent to each customer. Then consider the case of people who forget to pay, and have to have second notices sent to them.
What would happen if somebody never sent in their check for the fee? The service provider would eventually pull the money out of the account. they wouldn't close their account.
Note that this same thing is done with health savings accounts, and I imagine 529 plans. The money used to administer the plan is pulled from either the plan balance, or the service provider gets their fee by holding down investment returns. I know that with the HSAs the money pulled from account is specifically mentioned as being an allowed expenditure that is non-medical but doesn't trigger taxes and penalties.