Franklin probably could have done it, he chose the poorest route that rarely thaws back to open water (in his era). He basically proves that it's unnavigable though by succeeding.
The knock-off effects are fairly minor and mostly Arctic related. His failed attempt probably garnered far more Arctic exploration than his success would have... there were several expeditions that went out in search of Franklin's expedition. These never happen if he succeeds and the area likely is explored only decades after it happened OTL without them needing to know what happened to him.
The greatest of the polar explorers of the era, John Rae, never has his career and legacy scuttled because he reported that a good and God fearing British crew broke down and resorted to cannibalism.
The surviving crew probably all die of lead poisoning several after their return. The lead concentration in their bodies was absurdly high, even beyond what the canned food could have gotten them (many suspect it had something to do with the way they stored and purified water).