Sorry, I'm claiming they would lose their primary source of funding in the sense that the next time the contract came up for bid no one would want it.

I'm at least somewhat skeptical here. The value gained by Google from all those searches is not exclusively in the ads shown. It might lower the value of the contract, but I would be at least somewhat surprised if it lowered it to zero.

There is value from seeing which result a user apparently likes best, which lets you train your search engine better. But Google already has a ton of this and the marginal contribution from Firefox users must be minimal. Instead, probably a different, smaller, search engine would probably pay. Though, even then, simply the cost of serving acceptable search results could be higher than the marginal value of the data?

Either way, this goes from a contract bringing in hundreds of millions to maybe tens at best?

There's value in being the one to serve personalized results, get people to sign up for accounts, cross-sell other services...