First, let's remind ourselves of what a raw file is. It is a set of single luminance values for each photosite (a/k/a pixel well or sensel) on the sensor. As such there is no color information to a raw file. Color is derived by comparing adjoining pixels that are filtered for one of three colors with a Bayer mask.¹ But just like putting a red filter in front of the lens when shooting black and white film didn't result in a monochromatic red photo, the Bayer mask in front of monochromatic pixels doesn't create color either. What it does is change the tonal value (how bright or how dark the luminance value of a particular color is recorded) of various colors by differing amounts. When the tonal values (gray intensities) of adjoining pixels filtered for the three different colors used in the Bayer mask are compared then colors may be interpolated from that information. This is the process we refer to as demosaicing. How much bias is given to red, green, and blue in the demosaicing process is what sets white/color balance.¹ The gamma correction and any additional shaping of the light response curves is what sets contrast.
¹ Just for the record, the three colors used in Bayer masks are not the same three colors emitted by our RGB displays, notwithstanding all of the cute little red-blue-green checkerboard diagrams on the internet. They're each, respectively, closer to being most transmissive to the wavelengths to which the three types of cones in the human retina are most responsive. Please see Why are Red, Green, and Blue the primary colors of light?
The B&W image you see on the back of the LCD screen of your camera is not "THE raw image". It is a preview image generated by the camera by applying the in camera settings to the raw data that results in the B&W jpeg preview image you view on the LCD. This preview image is appended to the raw file along with the data from the sensor and the EXIF information that contains the in-camera settings at the time the photo was shot. This is also what you are seeing for those first few seconds when you open the image file on your computer.
Any depiction of the data in a raw image file that you see on your screen is not "THE raw image", either. It's one of a near countless number of possible interpretations of the raw data contained in the raw file. What set of instructions are applied to convert the raw data to the JPEG-like image you see on your screen determines how it will look on your screen. This is why opening the same raw image file in one specific raw convertor produces an image that looks different from the same raw file opened with a different raw conversion application.
I was expecting Lightroom software to display the image at its best,
and not a "bad" interpretation of it. Why does the CR3 format exist,
if not to have the best quality and most precise and uniquely
interpreted? How can I work with it?
If you are saving your pictures in raw format when you shoot, when you do post processing you'll have the exact same information to work with no matter what Picture Style is selected at the time you shoot. Some applications may initially open the file using either the JPEG preview or by applying the in-camera settings active at the time the image was shot to the raw data. But you are free to change those settings, without any destructive data loss, to whatever else you want in post. Most third party raw image processing applications will apply their own default settings that affects how their interpretation of the raw data will look on your screen when you open the image. In general, they ignore many of the in camera settings and apply a generic group of settings that produce a color image with the saturation and contrast "punched up" a bit, just as most in-camera JPEG engines do when set to "Standard" or "Auto" picture styles.
Option A: Use a raw conversion application that automatically applies the in-camera settings at the time the photo was taken to the raw data in the .cr3 file.
Canon EOS cameras since mid-2005 incorporate Canon's "Picture Style" selection of various ways of processing a scene to produce a photo with certain general characteristics. Which Picture Style is selected does not affect the luminance values from each pixel in the raw data at all. Which picture style is selected does affect the way the raw data is interpreted to produce the preview image embedded in the raw image file.
When you select the Monochrome Picture Style you will see a B&W preview image on an EOS camera's LCD. The raw data saved to the memory card will still include the necessary information to process the images in color later with a raw editing application. WARNING - Be sure you are saving the raw data. If you only save the pictures as B&W JPEG images all color information will be discarded!
Canon's Digital Photo Professional will, by default, open an image in the same Picture Style as was selected when shot. All you have to do to change it is use the drop-down menu and select another Picture Style. You can even create a "recipe" for one image and then batch apply it to all of the images before beginning to work with them. (Note: there is a setting in the preferences menu of DPP that can be set to either show original image which uses the in-camera settings or show image with recipe applied which will apply any changes you have made using DPP when viewing images with the Quick Preview module. If you shoot using the Monochrome Picture Style but want to preview the images in color then select the later and batch apply a recipe preset that includes a color Picture Style before opening them in the Quick Preview module.)
Option B: Simulate the in-camera settings as closely as possible with Lightroom or other third party raw conversion applications.
With third party raw processing applications such a Adobe's Lightroom or Camera Raw, Apple's Aperture or Photos, PhaseOne's Capture One Pro, DxO Lab's OpticsPro, etc. getting images to display according to the in camera settings can be a bit trickier. Adobe products, for instance, totally ignore the maker notes section of a raw file's EXIF data where Canon stores information about the Picture Style selected at the time the photo was taken. Just how convoluted of a workaround is needed to accomplish such is outlined in the accepted answer to How to automatically apply a Lightroom Preset based on appropriate (Canon) Picture Style on import? In the present case of wanting to see the images in monochrome, you must select a different default profile with which to open the images from within LR's profile list that will allow you to develop those raw files from a B&W starting point.
For related questions here at Photography SE for further reading, please see:
RAW files store 3 colors per pixel, or only one?
Why are my RAW images already in colour if debayering is not done yet?
How to automatically apply a Lightroom Preset based on appropriate (Canon) Picture Style on import
Why can software correct white balance more accurately for RAW files than it can with JPEGs?
Why do RAW images look worse than JPEGs in editing programs?
The following is an excerpt from my answer to Why can software correct white balance more accurately for RAW files than it can with JPEGs? which you may find helpful:
Anytime you open a raw file and look at it on your screen, you are not viewing "THE raw file." ² You are viewing one among a near-countless number of possible interpretations of the data in the raw file. The raw data itself contains a single (monochrome) brightness value measure by each pixel well. With Bayer masked camera sensors (the vast majority of color digital cameras use Bayer filters) each pixel well has a color filter in front of it that is either 'red', 'green', or 'blue' (the actual 'colors' of the filters in most Bayer Masks are anywhere from a slightly yellowish-green to an orange-yellow for 'red", a slightly yellow green for 'green' and a bluish-violet for 'blue' - these colors more or less correspond to the center of sensitivity for the three types of cones in our retinas). For a more complete discussion of how we get color information out of the single brightness values measured at each pixel well, please see RAW files store 3 colors per pixel, or only one?
² Please see: Why are my RAW images already in colour if debayering is not done yet?