Having to tick the tick box every time is not an issue if the state is stored in some way, like with the "Enable auto-editor privileges for this edit." box. Even without that, trying to perform an advanced (or direct) search from the search field is even more annoying: You have to load a result page that does not contain your result AND tick the combo box AND re-submit. You can use custom browser searches or userscripts to make these search types less cumbersome to perform, but MBz itself provides nothing to make this easier.

Visual clutter might be a problem though, as the space in the top bar is rather limited.

Using the advanced search by default was already suggested in MBS-2684 and rejected. It's nothing to do with performance - the whole reason the default search exists is because the advanced search is not suitable (in its current form) for ordinary users, because it requires knowing the ins and outs of Lucene syntax (e.g. "p!nk" won't include the result people would expect, "ac/dc" isn't even a valid search, and you definitely can't just paste a catno, barcode or whatever). Personally I think there should just be one search which is smart enough to know that "ac/dc" is not a regex search but "type:group" is trying to search the "type" field for "group", but that's probably another ticket entirely.

I'm not in favour of a checkbox. I think the majority of people will either be people who rarely (if ever) use the advanced search, in which case having a checkbox on every page is largely unnecessary visual clutter, or people who prefer using the advanced search, in which case having to tick a checkbox every time is just annoying. Either way, that doesn't seem like a good solution to me.