On touchscreen computers, icloud.com will not work with mouse or trackpad, have to use touchscreen.
I have an Acer R11 Chromebook, which has a touchscreen. Whenever I access iCloud.com on it and go into the photos section, I am forced to use the touchscreen to navigate because using the mouse does not work, the whole webpage is unresponsive to cursors.
This is also unfortunately the case with Windows computers trying to access iCloud Photos via iCloud.com, not just ChromeOS.
Perhaps the website recognises the touchscreen and doesn't expect having mouse input as well? Either way, it certainly hinders use of the site with a touchscreen computer.