Welcome to Color Management 101! Your are essentially correct, although the problem is deeper than you describe.
The best way to look at it is as three sided, i.e. the profiles associated with the image, display, and printer, with the CMS (Color Management System) inside Photoshop as the traffic cop. The rendering space you choose within ACR will define the color gamut boundary of the image, with sRGB the smallest, Adobe1998 in the middle, and ProPhoto the largest (forgetting the fourth choice Colormatch since it is of no practical value to most of us). For this reason many of us choose ProPhoto in order to not limit the editing range.
However the display profile will usually limit the colors you can see regardless of the chosen image rendering space, since the gamut of most displays is less than sRGB. You can still measure the colors with the eyedropper, but those outside of the display gamut will be visually clipped to the closest available inside the gamut. A good example is the bright yellow of some flowers, even though you can measure color differences across a petal the display will render only one shade. For this reason if the image is destined only for email/web viewing sRGB is the best colorspace choice, since an image in ProPhoto space will look very flat and unsaturated on non-color managed systems.
There is a lot more to the subject, particularly when you factor in printing, since many printers will render colors outside of the display gamut, sometimes tough to visualize the final print from the display rendering. Many books exist on the subject, time to start Googling.
thanks for the reply