Assuming that the MTA and Veolia had the same financial facts, its amazing how each decided to solve the problem. The MTA wanted to eliminate service but Veolia wants to adjust service, without eliminations. In fact, Veolia wants to increase service on some routes.
From a financial standpoint, every company has fixed (indirect) and variable (direct) costs. Fixed costs are generally overhead (buildings, buses, and front office salaries and benefits) and variable costs are the direct costs of operating a trip (salary and benefits of the operator, gas, bus depreciation, etc.) Fixed costs are "fixed" because they don't change based on the level of service.
By the county stating it saved millions of dollars leaving the MTA, this suggests that the county was paying a fortune in MTA overhead and Veolia overhead is much less. Spreading overhead is a mathematical exercise in allocation. Possibly, none of the bridges and tunnel surplus was directed to covering LI Bus costs.