What a great question, thank you for asking!
In keeping in line with the rules of the sub, I'm going to try and stick to hard facts as "I was there and witnessed..." and "I think, therefore..." are discouraged. This is going to make my answer a little more difficult as a lot of this is going to be rooted in those. I'll stick to facts as much as possible, and if I have to qualify I'll clearly mark it as my opinion so you can take it or leave it.
The short answer is.... yes!. The slightly longer answer is .... yes, to put it lightly!
To begin, let's skip the immediate post disaster frenzy, not because it's not valid and important- but because it's what naturally follows major, world changing events and your question and my answer is geared more to if Titanic stayed around and why it continues to stay around. To be brief- the shock of the Titanic event paired with the melodrama of Edwardian society and mourning in general spurned a huge market of Titanic pop-culture. Books, songs, poems, music (so much music), souvenirs, a dance craze, Teddy Bears, people beating up movie theatre owners, false advertising of footage (a relatively new technology and something that didn't really exist of Titanic)- it just went on and on. Three Titanic movies in 1912 alone. Massive public funerals and memorial concerts and statues and... well, you get the picture. The world was shaken and obsessed with Titanic until 1914 when war broke out and people moved on to other things.
...which is common. Events live in the public conscience until the next thing comes around.
But then, Titanic stuck around. Let's look at some numbers between 1912 and 1997-
Films: Ten- three in the 50's alone. These are just movies- these aren't documentaries (countless), episodes of tv series (countless), or works referencing Titanic (ie: I didn't count " The Unsinkable Molly Brown" as a film about Titanic although she features heavily)
Theatre/plays/dance/opera/etc/radio: I know of two musicals and 4 plays off the top of my head- but there have been many. I know Titanic was a popular topic in blues music. I don't even know if such source exists to count pop culture references as they are ever coming and it would be impossible to track them all.
Books: There was a massive amount of books post sinking but then relatively quiet until the 50's when "A Night to Remember" was published and became a hit- which began a demand for more Titanic reading material. Around this time, the Titanic Historical Society was formed which began to host conventions and unite enthusiasts around the world.
Toys/Games: Titanic has always been a popular model for hobbyists. I know of two pre-97 video games but I don't really know anything about video games so there might be more.
James Cameron's film was an absolutely massive cultural phenom, once again reigniting the public's interest with Titanic, but it was just one more in 80 odd years of Titanic media. The 90's weren't even a quiet time for Titanic- both video games, a whole bunch of post discovery documentaries, IMAX, a made for TV movie all preempted his film.
What it did do is create one of those moments where the public suddenly becomes obsessed with Titanic again. The aforementioned "A Night to Remember" kicked off a uniting and revival of Titanic enthusiasm mid century, which ultimately spurned the attempts to find her in the 80's until her ultimate discovery made her front page news again- lasting until the film 11 years later.
Now, I'd argue that the film has rooted Titanic even harder into culture- we make documentaries every year that don't even have any new information, Lego just released a Titanic build, museums and artifact exhibits are permanent attractions throughout the world- but this isn't new. The Cameron film may have kicked it up a notch, and the more connected world it exists in may have allowed it to grow (I'd argue), but Titanic mania, or at the least- pop culture symbolism- wasn't at all a new thing. It ebbed and flowed throughout the decades but it never, truly, went away.
The natural progression here would be "why"- and that's a massive topic that can only come down to opinion and observation, so I'll leave that one alone.
For more information and sources- honestly? Wiki has some pretty good articles on Titanic in popular culture. I know that seems like the lazy reference, but I don't know if anyone has sat down to tackle this academically- or even if you can.
Hope this helps :)