Here's the state of things:
Search history will work on a brand account in the YouTube Roku app, using these steps:
1) Reset the app (Settings -> Reset app)
2) Go to upper left "Sign In"
3) "Sign in with your phone" -> "Try another way"
4) Do not go to the URL shown, instead go to youtube.com/activate and enter the code
5) Select the brand account you want to sign in with on Roku, not your main Google account
Under "Accounts" (upper left), you will see your brand account... on the right part of the screen, you should see a "fake" e-mail address that ends in pages.plusgoogle.com.
Search history is working properly with a brand account signed in to the YouTube Roku app in this way. Unfortunately, starting today, you are no longer able to view age restricted videos when signed in this way to a brand account in the YouTube Roku app, even if your brand account is age-verified.
If you want, you can do an "Add account" and this time, on step 5, select your primary Google account. This will automatically add your Primary Google ID/email, and all associated brand accounts to the Roku app. They will appear under your primary main Google e-mail address. When you select one of these brand accounts, it will have **no** e-mail address on the right part of the screen. This will not affect the "brand only" sign-in that you did above... that appears separately, and again, has a "fake" e-mail address that ends in pages.plusgoogle.com. You can view age restricted videos when using brand accounts that appear under your main Google e-mail. However, when signed in this way, there is still the unsolved, ongoing issue of search history not appearing and watch history not updating. You will see "top" searches in the app's "search" section ("flame" icons) instead of your personal search history ("clock" icons).
I started a thread on Google support, but I don't have any confidence that it will do any good.
Hopes this helps some people...