I always assumed that the LHD's would have anti-submarine helicopters aboard, otherwise they are going to require escorts for the job they are quite capable of doing themselves.
As has been said, the LHDs will always be escorted (even for benign scenarios with no threat, Navy has said it will always be escorted), so there is no need for the LHDs to embark Romeos. And while they could do it themselves if absolutely needed, doing so would take away from their ability to d their primary job - being LHDs.
Embarking Romeos would mean an extra airframe type that needs to be supported, impacting personnel numbers, spare parts stores etc. It would take up space in the hanger, meaning less space for other airframes (or light vehicles and other stores, not the hanger deck is also the light vehicle deck). It would impact deck cycles (noting the deck can only really be open for flight ops around 12 hours a day, otherwise the deck crew becomes fatigued. Keeping the Romeos on the escorts provides far more flexibility as online times can be rotated). Even simply embarking the weapons and other equipment would be in impost - storing torpedoes and hellfires etc.
Letting the escorts doing the escorting and the LHDs doing the LHDing means we get the best holistic capability for the amphib force.
Turning the LHDs into pseudo ASW/sea control carriers might be a latent possibility, but it would come at the expense of having a ready amphib capability. It would require a deliberate decision to be made at the very top (government level, basically) and it would take a long time (probably years) for the Navy to train up to a level of capability worthy of the investment of so much resources.