The "log in again to do something secure" is a rabbit hole, where suddenly you have to log in again to do anything it all. It could become a login nightmare.

Only if 'something secure' is defined overbroadly. Using Amazon, as an example, I can idle my session indefinitely, and when I return, it will even allow me to do potentially secure things like browse my order history, look at my account details and such, but before I order something, or change my password, it prompts for a login, except where I've just recently logged in.

There's a balance to be had, and while admittedly, getting the details right is tedium and minutiae, it can be done, and done right.

I expect people would not understand this. They would start to get confused between their Google password and that of the service they are trying to access.

Also, after users learn the new process, it opens up a new phishing attack vector for Gmail.