Aviation began in Alaska in 1924. By the winter of 1924-25, there was a grand total of three aircraft in Alaska: two Standard-Js owned by Alaska Aerial Transportation Company, and a de Havilland DH-4B owned by Farthest North Aviation Company (sole pilot: Carl Ben Eielson). The Standard-J had the Hall-Scott A-7 water-cooled inline engine (notoriously unreliable and prone to things like spontaneously catching fire). The DH-4B, updated for postal mail, had a Liberty L-12 water cooled engine.
All three aircraft had been decommissioned for the winter and had nowhere near the range; a direct flight from Anchorage to Nome is 875 km, and the longest flight had been Eielson flying the mail 450 km from Fairbanks to McGrath.
Their main issues were twofold: the first was that they would have to stop somewhere to refuel, and with aircraft being extraordinarily rare in the territory to that date there simply were not many places for them to refuel, aside from possibly McGrath, and McGrath to Nome is 500 kilometers direct. The second issue is that they had water-cooled engines. Water-cooled engine + having to stop to refuel and overnight in Arctic conditions (you did not fly in the dark in those days) = solid block of ice in the morning.
So, those planes didn't have the range, they didn't have any place to refuel, and even if they had a place to refuel, the engines would have frozen.
@DJohnM referenced Wop May's 1929 flight. While not taking anything away from May, his flight covered a total of about 640 kilometers. On the first night when May and Vic Horner stopped, they did so at McLennan, a village on Lake Kimiwan, where they could get shelter and refuel, then complete the flight the next day. So two things right off the bat: they had a significantly shorter distance to cover, and they had places to stop along the route, as they were following the Peace River and it had settlements and villages along it. That is, in fact, why their flight covered around 640 km instead of the direct 550 km from Edmonton to Fort Vermillion.
Bush pilots had been operating up into the Northwest Territories, thus even further north than Fort Vermillion since 1920 with flights between Edmonton and Fort Norman (now Tulita) flying workers into the newly-discovered oil field, a total distance of 1,400 km one-way direct (and significantly further in actual travel), which also happens to pass over the area of the 1929 flight. According to one paper, by 1928 there was no spot on the Canadian mainland that was considered inaccessible by air, with planes having flown to the Arctic Coast for various reasons, so by 1929 this part of the country had a group of pilots who knew the territory and had flown it in winter, which was not true of Alaska in the winter of 1924-25.
This might be hard to grasp it these days, but until the 1930s the United States lagged behind Canada in non-military aviation. In the 1930s Canada moved more air cargo than the rest of the world combined and the concept of the bush pilot was very much a Canadian innovation, long synonymous with Canadian-based pilots, who had a history of it predating any American version, so the idea that what in Canada could be considered challenging but doable would be utterly impossible in American territory. May was actually somewhat annoyed and bemused at the publicity and breathless reporting the flight generated, since he didn't think it was awe-inspiring as reports made it out to be, but did use the publicity to establish his own airline.
Flying had only started in Alaska that summer (1924), and no one had flown further west than McGrath. It was totally unknown territory to the two or three pilots available in Alaska, and they had infinitely more experience than trying to get someone up from the States, as literally no one else had flown in Alaska. And not even those two or three had any experience flying in the Arctic winter.
Also critically, May and Horner flew an Avro Avian, and the Avian had a Cirrus air-cooled engine. They didn't have nearly the concern with the danger of it freezing.
So, long story short, in January 1925 there were no pilots who knew the region to be able to fly into Nome, there were no places to stop on the way, and the equipment wasn't capable of operating in the winter conditions. By January 1929 northern Alberta had years of aviation operations, there were pilots who knew the area and had flown it during winter months, there were places they could stop along the way, and their equipment was more capable of handling the cold. In 1929, airlifting vaccine from Edmonton to Fort Vermillion was risky, but reasonably doable. In 1925 trying to do so from Fairbanks to Nome would have been suicidal.
There were certainly aircraft in existence that theoretically could have made flight, and with the right engines might have made the trip in winter (such some models of the Junkers F 13), but none of them were in Alaska and all the other limitations would still be in effect: no infrastructure, no pilots available who knew the area. So at best the book's claim the flight could have happened is theoretically possible, but practically speaking no, it was not realistic, and thus the assertion it was a deliberate choice not to fly is simply nonsense.