In fact it is easier to get bigger discounts if you are a mid-size customer than if you are a big customer. If you are a very big customer, then any discount given to you will directly affect the cloud business bottomline.
A public cloud inherently/structurally has more costs because it is built to be truly multi-tenant and has lots of elastic capacity.
A single tenant private cloud can cut out a lot of bells and whistles and can be tailored to the needs of the company's private use-case – this brings down the cost significantly.
If your compute/storage bill is greater than $10 million per year, then it is highly likely that for the same money you will get much higher compute/storage capacity in a private cloud. And that gives you more headroom for growth/elasticity etc.
If your bill is greater than $30 million per year, then you will likely save significantly as well as get other strategic advantages – especially if your infra scale continues to grow over next 2-5 years.