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I tested extensions using the browser Supermium (portable version) which is based upon Chromium (Googla's Chrome). Many extensions are available for other browsers, so you can search for the same extensions for availability for you browser of choice.
The button page feature is commonly called a speed dial. The speed dial extensions provide a way to have a page of buttons that link to web pages. There is no dial, and there are no phone calls being made. A better name would have been link buttons or URL tiles.
My preferences are that I can get square tiles with rounded corners, customizable screen and tile backgrounds with solid coloar, display of the tiles starting from the top left, sizing of the tiles to my liking, and have some kind of grouping feature that allows switching to other screens of tiles. I tested many extensions. There are more, but the method of Googla's Chrome extension listing is infinite scroll with no indication of how many there are to be seen, and so at some point I did not want to continue. These little reviews below mostly describe deviations from my preferences and also any other annoyances.
boof (OP) double-posted this 4 hours ago, 1 minute later[^][v]#1,451,282
I will begin with the extensions that I found more appealing than most.
Speed Dial 2 New tab --can't adjust the tiles squareness (stuck with rectangles) but the ratio can be played with a bit by adjusting some settings
Group Speed Dial --a couple of unneeded buttons at upper right, can't change default tile background color (have to do each tile separately), some features require a paid pro version
Easy Speed Dial --setting to show tiles as square still makes rectangles unless you also set Title Placement as attached, setting as icons makes squares though but with same icon for everything unless you set Show Favicons also but that does not work consistently, tiles are based upon use of bookmarks stored in your choice of bookmarks folder