"Warm" = "more energetic", in every sense of the word. In my own mind, I think of global climate change in a more simplistic way: global energy change. We humans are releasing, in short order, energies that took natural processes millions of years to convert to the form that we're liberating. I can't think of any other philosophy or activity done by humankind that wantonly, aggressively and cavalierly "blows the wad" without experiencing a terrible consequence as a result of upsetting the balance.
Matter can be subject to a chemical or physical process that liberates the energies that keep it together. Temperature is one form of energy that is liberated. It's how power is generated -- we used that heat energy to, usually, make another fluid move faster, and convert that to mechanical energy by way of a turbine, and convert that mechanical energy to electricity, by way of rotating magnets inducing electrical current in a conductor. That matter that we burn/react/oxidize doesn't convert wholly to energy, though. Some of the physical waste products go into the atmosphere, changing how the air itself itself reacts to various energetic inputs, such as sunlight or radiative heat from urban centers. We literally convert matter that took millions of years to make into energies and waste matter that were not there in those quantities for those same millions of years, and the simple, basic physics of it is that the result of that conversion is waste matter and energies that serve to either directly increase heat energy (a small part of human-induced climate change) or change the energy storage properties of the largest, by mass, energy batteries around: the atmosphere and the oceans.
So, yes, when there are stronger hurricanes and more destructive tornadoes outlasting their historical durations and exceeding their historical energies, there is little else but temperature that causes that to happen. And, yes, it goes the same way for winter.
This same climate change can have a chilling effect on winters, due to the changes in the propagation of that energy-as-temperature affecting the jet stream or deep ocean currents. The atmosphere and the ocean are huge, dynamic, swirling masses of differing energies, and the weather we experience is a result of how those masses of air and water move, said motion being a product of their energies (again, temperature). Wind itself, hurricanes, storms, blizzards, tornadoes, etc. are all caused by the interaction of large masses of air at differing densities (itself one effect of temperature). The energy imparted to those air masses comes, in no small part, from the ocean.
I think if more people equated "temperature" with "energy", the layman's summaries of climate change may reach a few more skeptics. At least, the ones who want to learn.