Some would, but it definitely is a bigger ask of people to have something injected into their bloodstream by a medical professional rather than to just spray something in their nose. I think it'd win over a substantial chunk of the vaccine-hesitant.
Have you spoken to any vaccine-hesitant? Almost none of them are vaccine-hesitant because they don't like the idea of a needle in their arm; a nasal delivery method would have almost no measurable effect on voluntary uptake.
It IS a more traditional vaccine, though, and less scary because of that. And this is not being pushed to the fullest extent of the law (and a bit above it I think, but nobody can really prove that).
I don't think they'd say it's because they don't like the idea of a needle in their arm, but for a lot of them, I think it's a significant contributing factor.