I don’t like that Chrome forces you to install an extension to modify the new tab (to such an extent). With Safari, I have it pointed at a local HTML file. That file contains a mini web app with my bookmarks. It has keyboard shortcuts, history and fuzzy search built into it.
For Firefox I have my own little extension that just loads a HTML page with more or less the same as what you're describing. I presume you can also do that with Chrome, but I don't use it so idk.
I don't really get why this isn't just a setting, but ah well. There are some extensions that do this out there, but I found all behaved wonky. I'm not sure if they're "doing it wrong" or because I have it easier as I don't need any settings and can just load a static HTML file. I'm not an expert on any of this.
And then in newtab.html you can just put anything. It has the clipboardRead permission for some JS code I have to add a "go" button if the clipboard contains an URL