Today’s Question: When you say, “In addition, you’ll want to ensure that the monitor display is properly calibrated…”, I assume you mean on your own monitor, since you may not have control over the receiving monitor? I guess having the embedded color profile and having your own monitor calibrated will be sufficient to control the display on the possibly uncalibrated receiving monitor, right?
Tim’s Quick Answer: No. If you are sending an image to someone with an uncalibrated display, the appearance of the image will be somewhat unpredictable. But embedding the sRGB color profile will help increase the chances of relatively accurate color for the image.
More Detail: Most digital monitor displays are reasonably accurate in terms of color, and to some extent accurate in terms of brightness, even if they are not calibrated. However, if the display is not calibrated the appearance of the image will be much less predictable and may be rather inaccurate.
The ideal solution would be to convince anyone you are sending an image to that they should obtain and use a display calibration tool, such as the Calibrite ColorChecker Display (https://timgrey.me/calibrite) that I recommend. Odds are, however, that a non-photographer is not likely to calibrate their display.
Most monitor displays tend to have behavior that is relatively close to the sRGB color profile. This is the primary reason I recommend using sRGB as the color profile you embed in images that will be shared on a digital display such as a monitor or projector.
With this approach, if the software being used to display the image supports color management, the color will be reasonably accurate even if the display is not calibrated. If color management is not being used there is a higher risk of an inaccurate image display, but in this type of context there isn’t much you can do to improve the accuracy of the image, since you don’t have any way of knowing the behavior of the specific display configuration being used by the recipient of the image.