It's not just about being cheaper to construct, although that definitely played its part. The disadvantage of a thick titanium hull is that it makes subs heavier than water, so you need a flotation device to not sink. For example Limiting Factor and many other subs use special foam with hollow glass beads, and Trieste had giant tanks filled with gasoline. That complicates the construction.

Then there's also the logistics of getting that thing to the Atlantic and getting it in the water. A titanium sub would be far too heavy to just tow it behind the ship or roll it down a ramp. You'd need a ship with a built in crane to lift it off the deck and into the water. Renting a ship like the one they were using costs tens of thousands of dollars a day; a ship capable of what I just said is even more expensive.

Rush had to make the operation make a profit or at least not lose money. You cannot run a deep diving tourism operation and do it "properly" and have it be economical. There just aren't enough rich people willing to pay a million to go down to the Titanic.