how to use set_select on a form to show the old value?

No. You're looking at the wrong place. I pointed to "form_dropdown" so you don't need to write the <option> yourself Inside, he will choose the current "selected" himself.
Second, no, you don't need to write selected. This is what the function outputs.

PHP Code:

        if ($input === null) {
            return ($default === true) ? ' selected="selected"' : '';
        } 


You have error in request set_select('request', $booking->status) Why 'request'?

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