LibreOffice Online has the search always available in the statusbar. https://www.libreoffice.org/assets/Uploads/LibreOffice-Online-Calc.png Also KWriter, Kate have the search in the satusbar (there the statusbar switch to search). This behaviour will show search always by default, cause statusbar is on by default and the UI is similar to LibreOffice Online.
I would suggest show the Find toolbar inside statusbar by Ctrl+F instead. And for it we should increase hight of statusbar
so not always visible, only if you activate Find toolbar, like kate and kwriter did. Would work for me too. As LOOL is also LibO I'd like to bring some consistency between LOOL and LibO and have find in statusbar like LOOL has is an good idea from LOOL.
Definitely a WF from my side (reopen if you disagree). LOOL is a website and the statusbar is presented in the browser. It has a predefined height that's way too small for inputs controls. KDE apps such as Kate don't have any statusbar and do the same as us: the F&R toolbar pops up at the bottom of the application. And last but not least, the statusbar's purpos is to show a status not being a space for controls (sidenote: same applies for main menu/toolbar and Gnome killed it).