I think the fact that only a couple browsers offer this option is telling in and of itself. While I apologize for the inconvenience, Brave offers a number of ways to view, organize or display your bookmarks – all of which are solutions to the issues you stated:
Here are your best options at this time:
Bookmarks Toolbar: While this does take up real estate, you can easily hide it using Ctrl + Shift + B (note that you can also change the specific shortcut key/combo used to do this in brave://settings/system/shortcuts). Alternatively, you can have it set to show only on the NTP and hide itself on all other tabs.
Bookmarks Sidebar: The bookmarks option on the sidebar is an easy way to display bookmarks and is also easily hidden. The sidebar itself can be hidden with a customizable keystroke (Ctrl + B) or set to only expand on hover. You can also set a bookmarks panel button to display in the toolbar which will expand it when clicked (brave://settings/appearance --> Customize your toolbar).
Additionally, you can add specific sites to the sidebar directly as well, allowing you to visit those sites straight from the sidebar, rather than having to view the panel. The sidebar, as stated before, is easily hidden and would prevent taking up any additional screen space if set to auto-hide.
New Tab Page Tiles (favorites): Up to 24 tiles can be displayed per page on the NTP, all of which can be arranged to your liking and allows you to open a new tab and click to view the bookmarked site in the same tab.