I suspect if this were to go into effect, that yes, after the first winter on year-round DST, public opinion about the shift will change dramatically in a negative direction.
No one has yet experienced DST in the winter, owing to the present practice of shifting to standard time over the winter months. The change in morning daylight in the winter months will be a big shocker if this is adopted.
You're probably right. People at moderate latitudes--say 35+ degrees--mostly love their long summer evenings and they have tons of light in the morning anyway. And many may think they're fine with their cold dark winter mornings being a bit colder and darker in exchange for not having a time shift (and more light, such as it is, on the other end). But they actually aren't.
Considering how dark it is in winter mornings, I'd rather us just keep year-round off of DST instead. I say this while living in the geographic center of the country, and certainly for some people who live further north, I'd doubt they'd want the colder and darker mornings.