If this were true, then we'd expect to see similar levels of antivax around the world, remaining at a constant rate. Let's look at some actual numbers.
A world survey of the question, "Do you agree that vaccines are important for children to have?" found 97% agreement in Somalia, where the effects of vaccination are more observable, and only 68% agreement in France, which has had them long enough to take them for granted. In Asia, we see 95% in China, 93% in Vietnam, 96% in the Philippines and Cambodia, compared to 62% in South Korea and only 48% in Japan (down from 82% in 2015). 92% in Mexico compared to 79% in the US (which was 87% in 2015)
What's going on here? Did 34% of Japanese people suddenly develop a fear of needles in the span of 10 years? In the Philippines, 100% of people agreed with that statement in 2015, which dipped all the way down to 68% in 2021, before recovering to 96% today! Weird that that phobia of needles spontaneously appeared out of nowhere, and everyone got over it just as suddenly.
Or, maybe it's not really about a fear of needles at all. Maybe that has more to do with the fact that the US actively spread vaccine misinformation in the Philippines to get people to distrust China's COVID vaccine. And if we can directly observe antivax attitudes generating in response to a targeted misinformation campaign, then maybe this whole thing has more to do with the spread of misinformation and conspiracy theories, rather than some innate phobia.