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All of the knobs and twiddles are available and users do have access
Strictly speaking true. If I got in the cockpit of a 747, I would have "access" to the controls. If I had designed the thing, I'd probably think that the way to operate the controls was obvious. Your mission, should you choose to accept it, is to allow people who are only qualified on 21st Century motor cars to fly your 747. Expecting users to type query strings and look up ID numbers for users and forums is not providing access to the controls in any meaningful sense. If @Dammit is still having trouble, imagine how horrible the user experience must be for somebody without his considerable geek credentials.
All of the knobs and twiddles are available and users do have access.
Why can a vBulletin offer an "advanced search" page and we can't? Because they only search 2 dimensions of data within very narrow parameters, and we search far more.
Even then, if we tried, it would be a weird page... because so many options only make sense in combination with other options. i.e. no point saying "events that between x and y" at the same time as saying "only show conversations" because the terms would preclude each other and result in no results.
Our search is less similar in presentation to a forum search, and more similar to the search on Amazon, you start with a vague term given some localised (i.e. within a forum) context, and we show the controls that make sense given that context to drill down further, i.e. a search for only events suddenly shows event time based controls: https://www.lfgss.com/search/?type=event
Everything is there, it's just not pretending to be something it isn't and couldn't be: vBulletin search. It's different, but it's not less capable, and when people get it they love it, but you're not going to get it if all you want is something that isn't possible on here... the vBulletin Advanced Search page.