Without the Olympic sinking, there would no need to refit the Titanic nor to modify the design of Gigantic. However - would the Titanic strike an iceberg on her maiden voyage, if she had departed as original planed in 1912? If so, perhaps without the refit she would go down. Hell, even the Gigantic would had problems to survive being torpedoed off Ireland during the Great War without it.
The other thing is, of course, is that had
Olympic gone down in the Solent I think it is likely that the maiden voyage of
Titanic would have been captained by Edward Smith; he had a tricky record despite his seniority and publicity, and I'm sure that had he had command of Titanic he would have found a way to mess that up too. It is true that her striking of the iceberg was largely down to weather and luck, but the fact it happened even without Smith is testament to the quality of the White Star Line's captains it seems.
Well, the unfortunate sinking of the Olympic was in a way a saving grace for both the Titanic and Gigantic. Parts of the wreckage was recovered and sent to Harland and Wolff in Belfast for analysis, and its showed the steel used in its construction was more brittle than originally thought - even though Olympic had just started operations a few months earlier. The relatively cold temperature of the North Atlantic Ocean may have accelerated its steel turning brittle. As such, both the Titanic and Gigantic ended up being delayed for one year as both ships had to be heavily modified with new steel structural parts of much better quality.
The financial cost of these efforts to White Star Line cannot be understated, however. The
Olympic and
Titanic incidents were disastrously expensive for the company. Given the initial cost of the class, the tragic loss of
Olympic (then pretty much the largest liner in the world) and the further work that had to be done on her sister ships, for
Titanic to run into trouble and narrowly survive was catastrophic for PR. I find it pretty tricky to envisage that the White Star Line survived as long as it did in our timeline, war or no war.
Titanic today is a hotel/museum ship at the port of Southampton, just like the original Queen Mary is a hotel/museum ship at Long Beach, California in the USA.
I still think it would have been nice to send it to Belfast, which was one of the options considered. It would have been a nice testament to the history of shipbuilding over there.
This didn't prevented the Lusitania Disaster of 1914 where 1400 passengers lost their lives.
I thought that the actual cause of the explosion was unconfirmed? I mean, with little evidence I find it hard to believe that a cruise ship just... explodes. But I don't want to dwell on conspiracy theories.