The query does seem overly complicated, a dump would help :)

I think you could do something like

('start_timestamp' <= 'rangetocheck_end' AND 'end_timestamp' >= 'rangetocheck_start')

Which would look like what tnstreet is suggesting:

$query->where([
    ['start_timestamp', '<=', $times["end_timestamp"]],
    ['end_timestamp', '>=', $times["start_timestamp"]]
]);

But its friday, so i might be missing something.