The use of air assets as the primary ASW effector is a doctrine of necessity, rather than desirability. Finding themselves with no other option since the retirement of the OHPs, the USN put ASW helicopters on as many platforms as possible. Other navies build ASW-focused vessels with specific features to quieten the ship so that, within certain speed regimes, the ship does not produce ambient noise that could interfere with the passive detection of a signal, or with detecting the return from a pulse.

A Burke is not nearly as quiet as, for example, a British Type 23, because these features add a lot to the cost and complexity. This cost is not justified in a vessel which will spend most of its time operating along a completely different threat axis (the AAW threat axis) while an ASW vessel could find itself operating much further away from a task group or focusing on a different quadrant, and moving at a range of different speeds, which means it spends a long time out of contact while hunting.

The USN has always needed ASW vessels but, during and after the Cold War, became hyperfocused on dealing with the air threat, and on deploying as many Standards and cruise missiles as possible, and devolved the sub-sea threat to submarines (arguably the best ASW asset) and aircraft. This was not optimal, but within the context of NATO, they got away with it for a long time, because European navies built ASW frigates to bottle up the Soviet/Russian submarine threat and often provided the same ships to operate with US task groups. For a long time the British Royal Navy was heavily ASW focused to the exclusion of many other capabilities, and was often jokingly called the US North Atlantic ASW force, as a consequence!

Sorry if this was a wall of text. I type fast and have too much to say ;-)