I use Chrome Beta as my stock browser and I had the same issue in v123.
This has now been solved in the latest Beta (v124) so I would expect the same when the stable version upgrades to this. However regardless of this I have a far more preferable option.
I can't speak for other third party launchers as I've only used Nova Prime pretty much since birth but using this you can change the image of any app or shortcut icon added to the screen.
For example I have a Chrome shortcut on screen for the Met Office rain radar map from their website. The image on the left showing this doesn't tell me very much so I changed the icon to a rainfall image I downloaded as shown on the right.
Same principal applies to the shortcuts to the settings I have for Do Not Disturb and Voicemail shown underneath (this feature is also available in the free version of Nova).
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