Now everything's fine. Thank you, Mike!

Here is the print I got with the IntelliCenter. The first two images are OK but the one with the zeroed borders doesn't have the cut marks. Why is that? Actually, the only reason why I need the transparent borders is because I need the cut marks so I could easily cut my photos with white mats.

You're right about the white crush. However, to me personally, it's easier to fix than to stand the overall print dullness. Anyway, my experience is not like yours. Maybe I'm doing something wrong but looking at my prints I don't mind it at all.

Returning to my main question with that borders. Yesterday I tried to follow your advice with mats but I got weired results. I added three images on the page: first - without any mats, second - with ordinary mats, third - with the same mats but zeroed borders. Then added the cut marks (by the way, is it possible to apply the cut marks only to specific images on the page, not to all of them?).  On my print I got the first two images with strange cutting marks and the the third image had no cutting marks around the borders at all.

I admit that what you wrote is a consensus among those who print. I had also used Relative RI in the past. However, somehow I tried to test the Absolute Colorimetric RI and I liked it much more! Actually, one can make sure that it is a better RI by turning on the Proof Colors option in Photoshop. It will be obvious that the Relative RI look worse in comparison.
The only drawback of usin AC RI that I see is that the whites sometimes become too white but it's very easy to correct. The changing of WB of the whites is so minor that it usually can't be seen at all.

Thank you very much! This method looks not as straightforward as the ordinary Borders but it's better than nothing.

As for the Absolute Intent, in my opinion this is one of the most important things about printing that I've discovered! In the Relevant Intent my images from Canon Pro-100 had always been a bit too dark and dull. In the Absolute Intent they look just geourgeous, almost like on my monitor (it is calibrated of course). Yes, the whites sometimes look not very neutral but one just need to find/create the right icc profile.
Some time ago I also found out that Canon also implies that the Absolute Intent is the best one:

"Absolute RI Algorithm... When used with basic 4-colour printers, this mode has a tendency to produce ‘choppy’ gradations that results in visible banding. However, with 11-ink printers such as the Canon imagePROGRAF PRO-500, the banding effect may not surface due to the printer’s extremely large colour gamut. Depending on the image and printer, photographers may find that the Absolute RI algorithm produces more colour accurate photos.
...
In summary, any photographer looking at producing fine art prints should understand their colour reproduction needs. If colour accuracy is important, it pays to invest in a printer with a very large colour gamut like the Canon imagePROGRAF PRO-500 and print in Absolute rendering mode. If colour accuracy is not the paramount concern, the Perceptual mode, when coupled with a competent photo printer, will produce visually satisfying results". https://snapshot.canon-asia.com/article/eng/introduction-to-fine-art-printing-part-3-colour-profiles-and-rendering-intents

When one prints with an Absolute Colorimetric rendering intent the whites on prints usually have a color cast. It's understandable. However, when one wants to print a photo with borders why do they get the same color cast? Is it possible to make the borders transparent without any ink on prints? If it's not possible to do in the current version I wish it will be possible in the future.