Redesign new tab menu page -> dropdown (#20203)

## Summary of the Pull Request
Renames the "New tab menu" page to "Dropdown menu".
Redesigns the page with the following notes:
- the preview is now on the right side so that it's always visible
- moves preview buttons into border with previewed new tab menu entries
- when the window narrows, removes the icons from the controls, then
changes layout to be vertical so that it all fits.

Aside from the main redesign changes above, a lot of the other changes
are to make sure that spacing and alignment looks ok.

## Validation Steps Performed
- [x] Wide view
- [x] Medium width view
- [x] Narrow view

## PR Checklist
Related to #17000