It failed in it's initial intended role as a carrier borne strike aircraft e.g. also due to the engines you pointed out. But it was conceptualized/designed from the very beginning as
a carrier based aircraft.
No, it was not. It was designed for the US and UK Air Forces as both a low level penetration bomber, and a deep penetration interdiction fighter. That is why the project was created, as an Air Force project. The Navy had their own similar project, the XF-10 Jaguar. However, with the cancellation of that project the research gained was then transferred to the F-111. The Navy still wanted a smaller and lighter aircraft, but SecDef McNamara insisted that instead they combine with the Air Force in the F-111 project.
Something the Navy never wanted to do, because it was known at the outset it could never be effectively converted into a carrier based aircraft.
So no, it was not "designed from the very beginning" as a carrier based aircraft. It was designed as a land based one, the carrier mission was added later.
Unlike the Rafael or the J-11, etc. that were conceptualized as tarmac based aircraft.
Which Rafael? The Dassault Rafale was intended from the very start to be a land based aircraft and a naval one. Hence, the two different models, the D and M. Just like the F-35 A and C. The C is not a "ground based fighter modified" for Naval use, it was always intended to be a naval aircraft. Same as the Rafale M. Co-designed, not modified after the fact.
And this was done because when it was being designed, they knew the Navy needed to retire the aging Super Etendard. That followed decades of French designs, where as they were developing a new fighter, at the same time they would specifically engineer a variant for carriers. Not "modified", but designed both a land based and naval version at the same time.
And if one looks at every single one of those, with only a single exception (Iraq wo borrowed 5 Super Etandards), the only purchasers were nations that had carriers. Like Argentina. Everybody else always bought the land versions only.
And once again, the PLAN does not use the J-11, that is a Chinese copy of the Su-27 made for the PLAAF. What the Navy uses is the J-15, a Chinese copy of the Su-33. They are not the same airplane at all, even though all can trace their lineage back to the Su-27. But neither the Su-33 nor the J-15 is an Su-27.
Because not every country on this planet can afford the luxury and national state debt of the USA - due to e.g. building specific aircraft for NAVY, and the Air-force.
France e.g had build the Etendard and then the Super-Etendard for it's fleet air-arm, shit expensive and aside from e.g. Argentina no buyers.
How many nations out there have carriers that mandate the need for a naval aircraft?
The answer is obvious, not damned many. At this time it is the US, China, France, India, Italy, Russia, Spain, Thailand, and the UK. Many other nations have helicopter carriers that can perform many of the same missions, if they were to decide to buy VTOL or STOVL (V/STOL) aircraft.
Which means they would have to buy the F-35B, or see if they can find some old Harriers or Yak-38s laying around somewhere. Because those were the only 3 V/STOL fighters to ever actually make it past the prototype stage and go into production.
And France like the US is really the only nation left that maintains only CATOBAR carriers. And as they also have a policy that almost every aircraft they use has to be French designed and built, when the Navy decides it needs a new fighter it pairs up with the Air Force to design both variants at the same time. But as the Navy does this less often, that ends up being around once every 3-4 land based models.
Oh, and the Super Etendard was not from the Etendard. It was an upgrade to the Etendard IV, an aircraft designed and built for the Navy. Which ironically was part of a program to build a fighter for the Navy and Air Force called the Etendard II. The Air Force bailed out, leaving only the Navy. Then later the Air Force tried again with the Etendard VI, which was ultimately also rejected. SO once again, it was never intended for anybody but the Navy. And Argentina only went with the SE because they had no choice. They were using the A-4, until the US Arms Embargo during the "Dirty War" meant an end to their replacement parts. They then wanted to go to the Harrier, but their conflict with the UK left only France as a source of naval aircraft.
One thing that France had been doing since the Jet Age started, was having their Air Force and Navy work together for new aircraft acquisition. Unlike in the US, where that was never done until the F-35. All the other attempts were in adapting a land based aircraft, which ultimately failed. In France, a Naval variant was a consideration at the start. For the US, it was always something added after the initial design was made and they tried to then adapt it and it never worked.
And the Soviets like the French operated the same way. Not an "adaptation", but two different aircraft intended from the very start.
And an interesting side note, the Soviets once even planned on building a carrier that conventional fighters could operate from. Project 1160 was intended to operate with conventional VSR fighters like the MiG-23 and Su-24, as the variable wings gave them a much shorter takeoff distance without the need for a catapult launch. But when that carrier project was killed by the collapse of the Soviet Union, they had to return to the drawing board and update the Su-27K into the Su-33.