I am a bit unclear of your actual question but am going to try to answer in context.
There are, a bit over-simplifying, two types of monitors out there: Wide and narrow. Narrow (or just normal) are close to but never quite sRGB.
When you calibrate a wide gamut monitor, you can make choices whether to calibrate it to a subset space, e.g. you can calibrate it to be sRGB (or several others). You can also calibrate it to "native" meaning as wide as it can display.
So the term "Calibrate to sRGB" really has meaning only for a wide gamut monitor, but more generally could apply to either native on narrow (as most are close to sRGB) or to that target on wide.
When you run "calibration" software, there are usually two events, a calibration of the monitor, and a profiling of the monitor. Usually they happen together, but they serve two purposes. Calibration is where something in your system receives a transformation matrix, usually called a LUT, which says "when this RGB value is requested display that one". It may include simply lookup values, or more complex transformations (you may see terms like 2D and 3D for example). On many computers this transformation is loaded into the driver for the video card, on some monitors it is loaded into the monitor itself (so called "hardware LUT's").
This calibration provides corrections so that your monitor is more accurate. For example, if you had a line of monitors and all calibrated to the same target, they would be more similar to each other for a given image displayed. The calibration, however, has several somewhat subjective decisions you make in addition to the (if wide) the target space, such as the color temperature, contrast, black point. These tend to be driven in part by environment (color temperature) and hardware (contrast is limited by that). So it is not necessarily true that two monitors that are "calibrated" are calibrated the same, even if to the same target space.
Now... during "calibration" there is also usually a profile phase which will measure the monitor's actual response and provide a profile to describe the monitor's actual display capabilities. This profile speaks to the color space, or more precisely to how close that monitor got to the color space. Of paramount importance if you calibrate to a wide color space, it tells color management applications which one.
So essentially the calibration gets the monitor as near a target as possible, and the profile is information to describe how the monitor was calibrated, and how well it responds.
In the days where everything was approximately sRGB, then a "calibrated" monitor was simply one corrected to be more accurate, and color management was (relatively speaking) less important since most things were sRGB and most software was not color managed.
With more wide gamut images used in editing, and/or more wide gamut monitors, now we need to translate. If you try to display an sRGB image, an AdobeRGB image, and a ProPhoto RGB image by simply sending the RGB values to the monitor, you get three very different images (regardless of the monitor's calibration, though which color space would imply which one was best). A specific color's numeric value in each color space is different.
The "calibration" produces a profile that allows color managed software to map one into the other, so a monitor calibrated near to AdobeRGB, can display an image in ProPhoto RGB by using the profile to translate the RGB values from one space to the next. Thus a color managed application displaying the same three images should display the same for all three, regardless of monitor type (so long as it is calibrated).
Note that this does NOT Mean it will look the same on a wide monitor as a narrow; there are colors one can display the other cannot. But it will be as close as practical, and more importantly consistent between images even in different color spaces for those colors actually in those spaces.
You mention the controls on the monitor - when you calibrate the monitor, it is useful to change those in certain ways (the calibration software will usually help), but that is not the driving force behind the calibration, it is the LUT table loaded. The driver will adjust both color managed, and non-color-managed applications based on the calibration, regardless of the adjustments. However, adjusting the monitor after calibration will (on most systems) break the calibration -- you will have essentially given the calibrated setup a different monitor.
Bottom line: Calibration makes the monitors more accurate to their targeted parameters; the profile allows color managed applications (and only those) to translate between color spaces (including the color space of the monitor). It takes both to be complete. But calibration to sRGB (or on a narrow monitor) means sRGB images in NON-color managed applications appear more normal. And there are more sRGB images on the web than anything else, hence I believe the genesis of your advice.