"Shouldn't we consider it a giant, massive red flag that you need a corporation backing you to maintain one of the most critical pieces of web software?"

Yes.

Interestingly Mozilla Foundation is not asking for donations.

Mozilla Corporation is selling traffic to ad-supported search engines, and profiting handsomely.

Instead they are asking for user cooperation in their experiments.

Why not create a different breed of browser that does not expose users to advertising. Profit motive?

Let users support it with donations.

This should be the mission.