After watching all chapters of the series and reading all reviews, in my opinion I do think Merlin and Arthur both fulfilled their destinies.
The producers did an amazing job showing their destinies and producing a good closure of all 5 seasons.
Here is how I understood the ending and the destinies of both Merlin and Arthur.
Albion: the world which Arthur and Merlin created
Throughout the whole series since the beginning, we heard of the wonderful unified lands of Albion and that magic will finally be allowed and respected. What we didn't realize, it actually has all been done and fulfilled.
Since the start of the series we started to imagine a world of King Arthur with Merlin besides him not hiding his magic and the use of magic in the good way throughout the unified kingdom.
The fact Arthur died in the final chapter doesn't necessarily mean that both Merlin and Arthur had failed.
Both actually succeeded, Arthur's destiny was to unify Albion which happened.
Most enemies became allies, Camelot's sole enemy was Morgana and her army towards the end. The war was won by Arthur's kingdom meaning Arthur has succeeded to unify all lands. Just before the final battle King Arthur talks about the unified lands.
For as Merlin, his destiny was to protect Arthur till Arthur could fulfill his own destiny and that was Merlin's destiny; to fight in the final battle as his truly whole self. Merlin's power was at his most powerful in the battle and with Merlin's help there was victory to Camelot's kingdom. With some help from his father Balimor, Merlin was able to believe in himself and use every extent of his powers in that battle.
Truly, anything else could have happened and many what ifs but that's the story the producers wanted to tell.
Post King Arthur's death
So we've all waited for that 'paradise' called Albion, good magic will return and all magical people and creatures will be respected.
Even if Arthur was to live after the battle, he would know Merlin's secret of sorecery. Due to the fact that Merlin's sorecery saved Camelot, it would be obvious that King Arthur will allow good magic in the kingdom.
Alas the fact King Arthur died doesn't necessarily mean it won't happen. Queen Guenivere who is ruling Camelot after King Arthur's death, understood from Gaius that the sorcerer in the battlefield who saved Camelot was actually Merlin. Thus she would allow good magic back in the kingdom and probably upgrade Merlin as her advisor or as the Court's physicist with Gaius.
Therefore, the time that all sorcerers and priests have been longing for, all prophecies and poems written about that era will come true. Eventhough it's Guenivere's rule but it's still King Arthur's reign.
As a conclusion, in my opinion both destinies have been fulfilled; Arthur's and Merlin's. Even The Great Dragon said when both Arthur and Merlin were by the lake of Avalon in the final chapter "all has been passed, you haven't failed". Also even Balimor in the cave told Merlin while Merlin was going towards the light before the battle of Camlann "Go fulfill your destiny".
Probably we have been imagining the end abit different by King Arthur growing old and Merlin helping him around with magic.
Also probably because the Great Dragon said in the first chapters about Mordred's destiny: "If the druid boy lives, you can't fullfil your destiny". What we probably didn't realize and forgot that the fact Morgana and Mordred were to unify in evil was avoided by Merlin when Mordred was still a boy.
Probably if Merlin didn't stop the unifing forces of Morgana and Mordred at that time it could have been worse than what actually happened and Arthur's death would have been before the final war of Camlann. Probably we all had in mind if Mordred's alive the destinies won't be fulfilled which wasn't true. Modred did kill King Arthur eventually being heartbroken but it happened after Arthur fulfilled his destiny of unifing all lands of Albion which allowed Merlin to fulfill his in full power.
Thanks for reading. Eleanor B.Y.