Gavi has the right skillset, strong character and talent to be a longterm starter at Barca. But he must improve many things. Hardly surprising of course, he just turned 18. His career at barca will depend on his improvements in the near future. But he has more than enough time to improve. Iniesta for example was around 22-23, when he became an undisputed starter.
Frenkie can't play Busquet's position in our system. Frenkie loves to take the ball and carry it forward. But you can only do that with a double pivot, when you have a player next to you who secures the space you leave. That's where Frenkie has his biggest strength. If you remember the scene a few years back with Ajax against Madrid and Modric. This following scene shows his biggest strength perfectly. He has the physique to fight, win the ball and then carry it forward.
But Busquets plays it completely different. Sergio is the link between the defenders and interiors/flanks. He doesn't carry the ball forward, but passes it. Busquet's job as the only pivot is to open a play with a good pass forward, because he can't leave the space as the only player infront of the defenders. He needs to see the opportunity to pass and start an attack, while being under pressure.
Both players interpret the same position completely different, because of different skillsets. That's why Frenkie can never be Busquet's successor and I believe Xavi sees it the same way, otherwise he would have let him play there way more often.