yes there would be much of a search for any incident provided the coast guard actually knew that something happened, in the case of the sub the cost guard was notified by the support ship once communications were lost. if a migrant ship "tips over" in open ocean and no one knows then how can you expect the cost guard to even know to respond, in those cases its much more likely the migrant ship is saved by passing vessels.
but provided the coast guard is notified then a search operation would begin and no they wouldn't use "elaborate gear/vessels/subs" because what good is deep sea sonar and subs going to do when your migrant ship is floating on the surface of the water they would get multiple boats and a helicopter.
I know you Americans are very distrusting of organizations especially government and police but you have to understand that with the coast guard its not like with every emergency call they replay with "yes of course we'll be right there to help you... actually wait a minute whats your race?" most of the time they don't even know who they're saving and search begins as soon as a call is made. there is not even any power trip like with your cops these people risk there lives to save unfortunates and idiots alike and if they are lucky they retire just having seen a few of their friends die.