It's worth noting that starting in VS 2019 v16.4, you can now have "vertical tabs" which will list all open tabs vertically, making it easy to see all open files.
In Visual Studio 2022, this issue now has a native solution. Go to Tools -> Options -> Environment -> Tabs and Windows -> Show tabs in multiple rows. Excess tabs will be shown on additional rows. There doesn't seem to be a limit on how many rows this can use.
Another relevant option that some may find useful; you can choose "Show invisible tabs in italic in tab dropdown menu". The dropdown button is only visible when you hover your mouse near the end of the row of tabs. Showing invisible windows in italics makes it easier to identify windows that are only visible in the overflow menu.