Always show 'Configure Modes' experience

[pierceboggan]

The current modes experience is not discoverable. We should always show the Manage Modes dropdown, similar to how we show Manage Models for bring your own key.

Additionally, should we align on Manage vs Configure terminology across BYOK and custom modes?

[roblourens]

Will do this before shipping once it feels stable. I think I'd just like #252878 fixed