Upon actually reading their thread, I found that they found a solution in our forums! Apparently this can be caused by misidentification of the locale of InDesign in macOS. I don't know if it will work for you, but it's certainly worth trying.
Incredible, but it works!
I think this is what happened, if anyone has the patience to read it.
I normally use macOS in English locale, even if I am Italian and based in Italy. At a certain point during April 2025, when Apple Intelligence became available in Italian, I switched my macOS locale to Italian. This broke the Search bar in InDesign and in Steinberg Dorico.
In macOS Sequoia 15.4.1, going to System Settings > General > Language & Zone > Applications and adding both InDesign and Dorico to the list while specifying the English language made the search bar reappear.
Dorico's UI language can be changed to several other languages from the app's Preferences. Setting it to Italian, with the macOS locale to Italian, and the app specified to English (impossible to specify it to the macOS default) doesn't show the search bar. I will report it to Steinberg as well.
It seems that different apps within Adobe CC are localised in different ways? That's peculiar, since a coherent method would surely avoid this kind of issues.
Thank you for your help once again!