I want to know if the car leasing company has any responsibility on notifying customer on outstanding balance
This matter is uncertain, but you might want to assess it together with the related issues of good faith & fair dealing as well as the company's apparent invited error. More specifically, you need to ascertain from both the contract and the terms of the automated payment system the following items:
Whether the contract provides that company has the obligation to which your question refers;
whether the explanation given by the customer service representative is premised on the contract; and
if the previous item is answered in the affirmative, whether the automated payment system was supposed to withdraw a fixed amount instead of the current balance.
Most likely the company will posit that it is commonplace practice to bill for the period that initiates (similar to how it happens with leasing of real estate). Hence the importance of ascertaining item #2, since the contract would supersede a commonplace practice. However, from what I gather, the company's conduct sounds in invitation of error insofar as it systematically turned a blind eye on the fact that you at all times made payments as scheduled.
If the payment system was set up to withdraw the current balance, the company's invitation of error would evidence either a deliberate [company's] scheme to essentially bill in excess of the agreed amount, or a flaw in the company's billing system. Consequently, you would be entitled to a refund of late fees.
The matter is more complicated if the payment system was set up to withdraw a fixed amount. In that scenario, you would need to focus on the argument that the [contract law] implied covenant of good faith and fair dealing should have prompted the company to alert you of the situation instead of permitting a systematic re-occurrence thereof.
Lastly, the reason of being of late fees is, in part, to deter instances that could worsen and develop into customer's default. Here, since your long pattern of scheduled & timely payments essentially rules out said risk of default, the circumstances seemingly support a finding of [company's] unjust enrichment.