If you open the Photos app (macOS) connected to iCloud with an empty library, there will be no photos until you import them from your phone. Hope this is clear now.
iCloud via browser has a limit of 1k photos per download.
My bad — it appears that Monterey wasn't syncing with iCloud due to "Performance." Photos on Monterey hides the status message at the bottom of the page, and I had to pull the page down twice to finally see it.
iCloud via browser has a limit of 1k photos per download.