Hello!
I recently worked on a project with a similar task. My query for checking overlappings was big so I thought how to reduce it. After some time of analysis it became very simple, and lgical.

User::whereNotHas('availabilities', function($q)use($selected_date_time)
                                   {
                                    $q->where('unavailable_start_date', '<', $selected_date_time)
                                    ->where('unavailable_end_date', '>', $selected_date_time)
                                    ;
                                   })
                            ->with('availabilities')
                                                        ->get();

I'm not sure if there is a whereNotHas method, but it would work. Your query would return all users except those who have all availabilities unavailable.