I think it's worth remembering that the Last Alliance besieged the forces of Sauron in Mordor for seven years before he himself came out in a last ditch attempt. I doubt that the movie portrayed this at all accurately, as it would appear (from the book) that Sauron let his forces fight virtually to the last orc before he came out. So at this point what could Sauron, even with the Ring, do by himself against a huge force of Elves and Men?
Of course, in taking Sauron down both Elendil and Gil-galad died, but I'd put that down more to Sauron's abilities as a Maiar rather than to any power of the Ring. It seems to me that the Ring was designed to give Sauron power by controlling the other Ringbearers, who were the leading Elves, Dwarves and Men. The Ring was not designed to make Sauron into some kind of unstoppable killing machine. The Ring was partially successful in that it did give Sauron nine mighty wraiths - but it didn't work on Elves or Dwarves.
It might also be worth remembering that at the time of the Last Alliance, Sauron had already died once - in the downfall of Numenor - so that he may not have been as physically powerful as he once was. He may have used much of his strength in creating a new body - which he had done fairly quickly.
All in all - Sauron was defeated and/or killed five times after the creation of the Ring. Four of those times occurred while he was in possession of the Ring.
1. By the Númenóreans and Elves in the War of the Elves and Sauron (S.A 1693-1701). Sauron was forced to retreat after the Battle of the Gwathló, and so lost the war. I wonder why the Númenóreans did not pursue him at this point, when Sauron's armies were routed.
2. By the host of Ar-Pharazôn (S.A. 3255). Sauron surrendered to Ar-Pharazôn when his forces deserted him, but what else could he do? The Númenórean army would have eventually overwhelmed him and Ar-Pharazôn might have ended up with the Ring. Sauron wouldn't have wanted to risk that.
3. The Downfall of Númenor (S.A. 3319). Sauron died as a result of the actions of the Valar and Eru. Since everybody in the vicinity was also dead, Sauron was able to take the Ring back to Middle-Earth, even though he was presumably in some kind of disembodied spirit form. Or maybe he animated what was left of his lifeless body to carry the Ring back ... not pretty but it would get the job done.
4. The Last Alliance of Elves and (Númenórean) Men (S.A. 3434–3441). We know that this force was not as great as the host of Ar-Pharazôn, because Sauron's forces simply deserted him on that earlier occasion. Elendil and Gil-galad physically killed Sauron's body and then Isildur removed the Ring before the spirit of Sauron could claim it. Also, since Sauron had already died just a few years earlier in the Downfall of Númenor, this second death (perhaps before he was back to his "full strength") may have weakened his spirit to the point where it was virtually not functioning. It took Sauron many long years to "get himself together" and for an extended period of time he was not aware of the continued existence of the Ring. Presumably when the Ring lay undiscovered on the floor of the Anduin it was in a dormant or quiescent state. Once Gollum had the Ring, Sauron must have become aware of it; in any case, Sauron would have realised that the continued existence of the Ringwraiths meant that the Nine Rings still "worked" and that therefore the One Ring must be out there "somewhere".
5. Gollum falling into the chasm in Mount Doom, thereby inadvertently destroying the Ring which had just recently come into his possession (along with the previous bearer's finger). (T.A. 3019).
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