I think part of the problem with Peak Oil has been applying the behavior of American oil production to the world. In America, we expanded oil production fairly quickly, and by the late 60s/early 70s, had exploited all of the easily available oil. Other countries otoh, still had easily accessible reserves, so while American oil production declined because the price of oil didn't support continued extraction, this wasn't some function of geology, but instead of a function of the oil markets.
Currently, there is no untapped low cost alternative to world oil, so instead of seeing stable prices and the same Hubbert Peak, we'll likely see rising prices and a long plateau. Other factors that contribute to this plateau are increased efficiency of use (higher fuel economy in cars) and higher than anticipated PEDs (price elasticities of demand) in OECD countries. The refinery problems in the US were a great example of this. Higher gas prices actually put a substantial dent in petroleum prices because consumers cut back significantly on top of the reduction in refinery throughput.
Oil's use as various fuels also helps out with stable production because we can generally substitute any hybrocarbon for light sweet crude, and still get gas, diesel, kerosene, and so on. This is why oil refinery output can remain consistent even while crude output can decline, usually due to substituting NGLs for crude. Biofuels can also substitute for refinery products, but they're usually too expensive to bother with unless we have some kind of mandate specifying their use.
The only things I can think of that would result in a Hubbert curve with world production are lower cost alternatives (EVs, which may happen over the next decade or two), world wide consensus on limiting GHG emissions (Less likely than EVs), or an extremely large recession.