I don't like this statement:
That item carried him to his current ELO.
Unless that item is ardent censer during the ardent censer meta, no it probably hasn't. There hasn't been a single item as strong as ardent censer by itself for as long as I can remember. An off build like banner and zzrot portal on Garen mid isn't going to boost your rank, for example. Something like AP Ezreal isn't going to boost your rank unless it becomes meta. Usually these builds can be replaced by another item, multiple items or champion.
Most off-meta builds are found at a lower rank, you can usually find evidence by looking at the top of the ladder in major servers. Granted, there are exceptions (for example rat irl is known for his Twitch, and jungles with him too. Pink Ward, while usually found lower on the ladder, one tricks Shaco top) but usually they are uncommon. On the opposite side of challenger is bronze, which is where weird builds are more common. Youtubers like Kshaway and thelolhounds sometimes put players like these on their videos. For example, in a recently uploaded thelolhounds video part of his bronze spectates series, I saw a Shaco with a bloodthirster and a death's dance. Obviously that build is garbage.
An Amumu who buys RoA would probably have a higher winrate with a full tank build, a liandries instead of RoA or even a full AP build.
It's not going to be the reason you lose that game.
Maybe not, but it's possible. A Sona with lich bane can make a game much more different than a game in which the Sona has ardent censer. The same can happen when a tank jungler has an adaptive helm instead of a locket, or a Vayne against 2 reliable CCs who buys a zeal instead of a QSS during mid game. Ninja tabi, arguably one of the most broken items is able to completely warp the state of a lane, an assassin's power level, etc. If a Yasuo is able to buy an early ninja tabi against Zed, the lane is over, and this can turn the game into a win if Yasuo is good at the game. An ADC with a ninja tabi against something like Camille Kha'Zix reduces a ton of their damage, and if the support has one too then they lose a lot of their threat. A RoA at 10 minutes is far more different than a RoA at 20 minutes on Amumu or for any AP character in the game. Power spikes are very important and buying certain items at certain times can completely warp the outcome of the game