OH my - - this thread touches so many issues and I have a multitude of opinions on all of them. Please regard the following post as just that: my opinions and thoughts concerning your quest for the best city for weather in a tax free state. Really, this is four posts in one (what a bargain!) so I have numbered them.
(1) I can't imagine basing a choice of retirement location on only two criteria, no matter how important they are to you. I would suggest narrowing down the country to places that meet your top criteria to a reasonably satisfactory extent, and then looking at those places with all of your criteria in mind (not just taxes and weather).
(2) I think it would make much more sense to look at overall cost of living, than to look only at states with no income tax. Even if you plan to have a huge taxable income in retirement, your overall cost of living could make a much greater difference to your personal finances than whether or not some of those costs went to state income taxes. I would suggest a more in-depth cost of living approach.
(3) As to weather, I think you might want to consider what is appealing to you about living in good weather. Do you plan to spend much of your time outdoors hiking, surfing, or something of that nature? Then you need to have those activities available at your ER location. If you don't plan to do much outside, then consider the implications of that accordingly.
(4) There is no perfect ER location. We all have to compromise, based on how important our criteria are to us. Also, there is the intangible. We narrowed down the entire world to three towns but when visiting them found that factors we had not considered, such as how the town was laid out, traffic, how hilly the town was, and whether or not we felt at home there were more important to us than we realized.