Set default status in CP admin for queried entries from "All" to "Live"
After a little head-scratching, the only way I could get this working was to register an AssetBundle (to load in my own JS), and making use of the Garnish toolkit as follows:

$(document).ready(function() {
    var statusSet = false;

    Craft.elementIndex.on('updateElements selectSource', function(data) {
        if (!statusSet) {
            // Find the option (and make sure it actually exists)
            $option = data.target.statusMenu.$options.filter('a[data-status="live"]:first');

            if ($option.length) {
                data.target.setStatus($option.data('status'))
                statusSet = true;
            }
        }
    });
});


This should only trigger the setStatus() process once, even when navigating between sources (which will also be filtered to to the saved status state of "Live"), but allows the user the freedom to change the status again if they so choose.

I hope this helps :)