Some aspects of how the SU toolbars (aka "palettes") work on Mac are because SU on Mac is multi-document-interface (MDI) whereas the Windows version is SDI. Since multiple model windows can be open simultaneously in the same Mac SketchUp session, only the top-stripe toolbar is associated with a particular model window. The others all float free (they belong to the app rather than to any window). Equally important, they are not forced to the "top" of the viewing stack; they can overlay or be overlain by other things including the model window, inspector windows, and other toolbars.
So, yes, it is more challenging to set up your screen on Mac to avoid issues with the toolbars. The MDI aspect is a Mac thing, as are other basic Mac vs Windows notions of how a GUI should work (e.g. the single app menu at the top of the screen). But a lot of the SU problems are programming choices made long ago that Trimble seems reluctant to change (in contrast to the Windows version, whose toolbar system has changed several times). Most of us Mac SU users have given up griping about the difficulties and differences since nothing ever changes; we get used to coping with it as is.
As to your specifics:
-
So far as I know, "New", "Open" and "Save" are not included on any standard Mac toolbar. I have no idea why not, but they simply aren't there and aren't available to add to the top-stripe. Other than the top-stripe toolbar, SU on Mac does not even support creating or customizing your own toolbars (one could do it via Ruby code, but that's a different question). I frequently envy the gurus using Windows SU who can show a small custom toolbar with just the relevant tools when creating an animation to illustrate a procedure.
-
At least on my Mac, the "Use large toolbar buttons" setting affects only the free-floating toolbars, not the top-stripe, and it makes them unacceptably large. However, this may vary with the specific model of Mac and screen resolution.
-
The available "snap" behaviors of the toolbars on Mac are very limited. They will snap individually to the right or left edges of the screen, but not to the top and bottom edges. They also won't snap to each other in any direction and don't recognize the edges of the model window in any way. Combined with the inability to keep them reliably "on top", this means you have to carefully manage where the toolbars, model windows and inspector windows are placed else things will obscure each other. As Dave says, you may spend more effort toggling them on and off to maximize visibility of the model window.
Some of these gripes are reduced if you have a giant high-dpi screen (hence lots of real estate) or dual monitors. But many of them are just facts of life.