The corporation basically exists to make the Google Search Deal possible. And considering that the search deal pays more than donations to the Apache Foundation and the Mozilla Foundation combined, it looks like it was a smart move.
* Total yearly income for the Apache foundation according to https://www.apache.org/foundation/docs/FY2020AnnualReport.pd... was 2.2M
* Total yearly income for the Mozilla Foundation according to https://assets.mozilla.net/annualreport/2018/mozilla-2018-fo... was $27M, about half from "program service revenue" (which means it came from the Corporation).
* According to https://www.mozilla.org/en-US/foundation/annualreport/2018/ (different document, but notice that both are for year 2018), the Mozilla Corporation got $435M from royalties, subscriptions, and advertising. In other words, while the Foundation is not technically allowed to give the Corporation money, that doesn't seem to be a problem.
I wouldn't ask why the need for the corporation. I'd ask why the need for the foundation. Though I'm pretty sure I know the answer: the foundation exists to limit the exploitative behaviour of the corporation. Remember: Benefit Corporations didn't exist at the time.
To quote their original documentation at https://www-archive.mozilla.org/reorganization/#q2
> By forming a commercial subsidiary, the revenue-generating activities of the new entity can provide funds to support development, testing, and productization of the various Mozilla open source technologies. This benefits both end-users of Firefox and Thunderbird, and developers and others who want to use the Mozilla open source code in various ways. Having the Mozilla Corporation handle revenue-generating activities associated with these products also allows the Mozilla Foundation to achieve its goals while still itself remaining a tax-exempt organization.