You can turn search suggestions on by flags. Paste link below and turn highlighted flag to “enabled”.(just “enabled”, without numbers) chrome://flags/#omnibox-on-device-head-suggestions-non-incognito
and if you want suggestions in private windows too: chrome://flags/#omnibox-on-device-head-suggestions-incognito
Sorry, you can just go to brave://settings/privacy and turn on "Autocomplete searches and URLs"
I recommend you to RESET FLAGS (go to chrome://flags and click “reset all”) because they can cause problems.
In recent versions, you were able to pin top sites, but they removed this feature. I use bookmarks bar instead of quick urls.
You can use extensions which change your new tab page, like Homey or Infinite dashboard.