Elrond's ring possibly along with inherent power gave him influence over the realm that he lives. I wouldn't say it gave him power on the battlefield that was not their purpose. To augment ones own natural magic maybe.
Anor literally translates to Sun in Sindarin. The Flame of Anor is not in any way shape or form related to the said Flame of Anor, Or The Flame Imperishable used by Eru to give life. In his letters Tokien talks extensively about the Three Rings and all their given names Narya the Great, The Third Ring, Narya the Red and The Kindler but never has he ever connected it with the Flame of Anor.
The Sun that is seen in middle earth is the last light of one of the Two scared trees of Valinor that of Laurelin, before they were destroyed by Morgoth.
What Gandalf saying is by "wielder of the Flame of Anor" is that he is a servant of the West and the Lord that live there an emissary of the Valar and that whom they serve Eru. He is not however saying, look at me I have a magic ring. He is above that, his power is above that. Naryas power lay in kindling the hearts and spirits of those around for valour and great deeds. There is also evidence they helped ease the passage of time on Gandalf allowing him to go about his task with more vigour but this is not the Flame of Anor.
Initially according to the HOME books it read as 'I am the master of White Flame. The Red Fire cannot come this way' reading it this way doesn't refer to the ring as a THING its a stance white for good, red and later black for evil. In this light even after its evolution into the changed poetic "Flame of Anor" it is Gandalf announcing whom he serves and the power he gains from that, he is announcing that he serves the good side the Lord of the West.
(The original texts for this passage are found in The History of Middle-earth volume 7, X The Mines of Moria (2): The Bridge)
In the actual text said in The Lord of the Rings Gandalf says
"I am a servant of the Secret Fire*, wielder of the flame of Anor*2. You cannot pass. The dark fire*3 will not avail you, flame of Udun*4. Go back to the Shadow*5! You cannot pass."
* The Secret Fire is the Flame Imperishable the force used by Eru to give life and free will. As a servant of this he shows he is a servant of Eru
*2 Flame of Anor as I stated is seen to represent the power he wields as a representative of the Lord of the West and his status as a Maiar. It is not the power of Narya the ring that allows him to cause the bridge to break it is his own. Inherent power.
*3 Again in contrast to the Flame of Anor and good we are shown Dark Fire, a reference to the enemy and evil to be drawn into comparison.
*4 Udun was Morgoths oldest and greatest fortress in the north of the world, Morgoth was the Balrogs only true master.
*5 Shadow, again a reference to dark and evil in contrast.~
Apart from the annotations just made and the reference to past texts and evolutions into what it became the sheer power Gandalf displays cannot be attributed to the Ring. Whilst you are entitled to your opinion of course, I think I have made mine now. I will give it that Elronds ring allowed him more possiblity and influence to create the flood at the ford, but I do not think it would have aided in him a sword battle.
Much like the use of the Eagles the Rings bore by Gandalf and Elrond are there to get our heros out of otherwise difficult circumstances or to provide a solution not to be overused to the point of being crutches for our characters to lean upon. It is also worth noting that Gandalf never revealed that he bore Narya, not even to those who travelled with him until his departure at the grey havens. Indeed it seems that until then only Elrond, Galadriel, Cirdan and Saurman knew for sure that Gandalf carried the Kindler. I doubt Gandalf would have so openly announced what he carried when we know that Sauron had many spies and was an expert in torture and getting information even out of Gollum about the One Ring's whereabouts.