I’ve just resolved a cookie issue by asking our hosting provided to explicitly not cache a cookie.
We had code that deleted a cookie. The cookie should have been deleted. It was not.
Due to their caching, the cookie was not being removed when requested.

My specific questions would be:

What advantages are they getting by caching cookies?
Is this a meaningful improvement?
Why do no other hosting services (that I’m aware of) do this?
Is it worth me caching cookies on my other sites?

Here’s a link to the support page that clued us in to what was happening: https://wpengine.com/support/cache/