Why is worrying about a single web server going down more worrisome than some part of the Docker stack going down and causing the same issue?
They are part of the same app.
They should not have the same level of privilege.
The secrets in endpoint A's memory should not be visible to endpoint B and vice versa.
Containers increase assurance that this is so.
But if a single process has the single account on the database, how do you partition those permissions? Simply providing multiple logins won't help if you assume hostile code is in your process space.
On the other hand, if each service has its own login, then the database can enforce lowest authority for each. A compromise of one service isn't a game over scenario.
It's the difference between having a single account with the union of all permissions, or disjoint sets.