Would a cut without the old lady storyline work better?

Including the modern stuff adds two important things to the film:

1) It gives us a really good look at Rose after the fact, and shows up what a monumental effect the experience had on her, how it inspired her to fearlessly chuck her old life and successfully begin a new one.

And 2) It gave James Cameron a chance to mess around with deep-sea exploration equipment. Apparently he's really into that stuff, and funds marine biology. Good for him, getting the film's backers to pay for his hobby!

I think it helped open the movie to a more general audience in that it helped the audience relate to it at the time, seeing an opening set in modern times. We also got that amazing transition from the old sunken ship to the vision in her mind as the "period" part starts.

Then there are those moments at the end as she dreams. That probably brought all the tears in the cinema and made people come back to see it again.

Plus we got to see Bill Paxton!

As others have said, the modern prologue and epilogue allow the audience to see how Rose was changed by both her relationship with Jack and the experience of the sinking itself. She's become a feisty woman with a zest for life instead of a repressed, miserable trophy wife. The drabness of the modern setting also makes the 1912 main plot look all the more glamorous and fairy tale like in a weird way.

And then there's the three-handkerchief ending where Rose passes away and rejoins Jack in the afterlife. That last triumphant moment would be impossible without the modern scenes.

Ticketsplease, if you want an authentic retelling of the Titanic sinking, without any two time periods intermingling and certainly no fabricated love story between two fictional passengers, maybe you should check out:

A Night To Remember
https://www.imdb.com/title/tt0051994/?ref_=nv_sr_srsg_0

It was made closer to the time of the sinking and I consider it almost a documentary, if not a movie, but it is still as compelling.

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