The reason why Arda chose Real Madrid over Barcelona was the departure time. As far as I know Barcelona wanted to have him by 2024 but Arda wanted to leave this summer. Real Madrid guaranteed that to him so he chose them. Both options were incredible for him of course.
Real Madrid is smart, they guarantee to take Arda as soon as possible. 
Barcelona may consider that they need to monitor the development of this player for 1 season more at least. So, they can sign him in 2024.
To be honest it wouldn't have been bad if he had played another season in the turkish league (and UEL) to gain experience. At 18 and with almost exclusively turkish league experience, he could fail in La Liga and especially under the pressure under Real Madrid. He's definitely talented, but you also need a strong character and be able to handle all the pressure.
Arda won't be in the main squad, I assume he will play with Real Madrid Castilla. He must take more experience and adaptation on La Liga with Real Madrid Castilla. So, when he is 20 years old, he can join the main squad. I think you think too pessimistic about the future of Arda in Real Madrid.