Yes, great comments and questions. I'm no PhD candidate either,
and challenge to authority is one of the main tenets of science. So there's no problem there! I do have a meteorology degree/background, so I have a certain understanding of how these things work.
The solar activity described in these solar storms consists of high intensity bursts of electromagnetic radiation, but the bursts are of short duration. The sun does indeed drive the weather in the lower atmosphere (I didn't typo it this time!). But the energy flux that drives the weather takes a significant amount of time to have an effect, as it's driven by the conversion of this radiation to heat energy.
By way of example here, note that the most intense solar radiation in the Northern Hemisphere occurs on June 21, the summer solstice. That's when the sun's rays are most direct. But highest temperatures in the summer don't occur until a month or so later. There are "energy sinks", notably the oceans, which take a long time to heat up or cool down, that tend to even out the energy flux from the sun.
So the short answer about solar storms is, these high intensity but short duration events don't really provide enough additional energy to the atmosphere to significantly affect the weather as we experience it in the troposphere (lower atmosphere). It's sort of "smoothed out" over time, as the conversion from electromagnetic waves to heat energy has these buffers in the earth/ocean system.
What these "solar storms" really impact is all things electric. The sun's rays are electromagnetic radiation, and intense radiation disturbs electric fields. So things like transmissions to and from satellites, cell phones/towers, radio, and other weaker electromagnetic signals can easily be disturbed. And apparently, it can really wreak havoc with these systems when there's a super-intense solar storm, like the 1859 event.
There haven't been any of these, shall we say, "super solar storms" lately. The story you quoted talked about the potential for future events to disrupt our current system, which does depend on so much electronics. So with no recent solar super-events, obviously there can be no correlation with recent weather events.
Now, if you want to talk about increasing CO2 concentrations and the resulting potential for extreme weather events (occurring as predicted years ago by computer models), that's the real angle. That's called global climate change, and that's where we should be truly concerned. My two cents.