The response on sending rescue teams was high even before it got popular on the news. I remember reading some statements that "we sent everything we can" in the very first news.
The problem is that there's no immediate feedback when they go missing so the search areas are pointlessly large to even attempt SAR operations. Areas where there's lots of immigration traffic are always monitored, like the Greeks were monitoring the boat before it capsized (enough to get that aerial photo of the overcrowded decks).
The vast majority of Coast Guards aren't stretched to their limits, they're sitting there ready to launch SAR operations if anyone calls, all largely operating for free as a cheap way of supporting maritime trade. The USCG alone responds to tens of thousands of cases a year, rescuing thousands of people. Some corrupt nations skimp on their CGs but that wasn't really the case here.
The Titan was A, your average fishing boat or other vessel is B. And all those migrants are, and that pisses me off to an incredible degree, C.
Occasionally we can see here on HN stories about lost sailors being rescued after weeks lost in the water, or shipwrecked on some remote rock. The common theme for those stories is that there wasn't a bunch of ships that went looking for them for weeks. For every person who survived such event, many more died.
When there is a lot of media coverage you also tend to get more reaction by officials, which then generate even more media coverage. It is the same concept why a individual can create a story on HN and reach people at google/facebook/apple, while thousands of users can have an identical situation and never reach a single person from support. It not a fair system but its a very well understood phenomenon.