The shirt sweating issue is more of an equipment and technique issue than a fitness issue. What you wrote about sweating in harsher environments is absolutely true. When one gets to higher altitudes, sweating through one's clothes can become not only uncomfortable but also dangerous. In warmer environments, wearing just a t-shirt can lead to hypothermia due to 'flash cooling', as wind rapid evaporates water in the shirt and lowers its temperature. Try wearing a backpack up a steep grade then taking off the bag and exposing a sweaty back to the wind at the summit to experience this. In colder conditions, sweating into one's clothes can ruin all sorts of insulation systems, reducing their effectiveness. A down or synthetic 'puffy' jacket will wilt when wet, frequently so-called 'breathable' hard shells will stop breathing when they wet through. Such smaller issues like wet base layers can develop in to big problems on longer treks. One continual issue on long winter expeditions is the gradual buildup of ice inside sleeping bags due to human sweat and respiration, thus reducing their effectiveness and increasing their weight each day.
There are some solutions.
Equipment wise: In warmer but variable weather, opt for wool t-shirts rather than cotton or synthetic in order to prevent flash-cooling. I know this sounds really uncomfortably warm, but modern light-weight marino wool and marino blends are very comfortable. Why this works: synthetic shirts dry very quickly, thus contributing to the speed of the flash-cooling. Cotton shirts absorb a ton of water, thus contributing to the magnitude of flash-cooling. Wool shirts absorb less then cotton and most importantly feel dry even when wet due to little tiny irregularities and ridges on the fibers. This keeps the water off your skin and decreases the chances of flash-cooling. In colder conditions, the secret is that usually people wear too much clothing. When you start out moving, you should actually be a little cold. As your body starts working, the heat will bring you to a comfortable temperature. When you stop to rest, put clothes on (a puffy works great for this) to keep the heat, then strip down again when you start moving. Lastly, if you ever put on a non-breathable layer like a rain shell, take off some layers underneath. Shells are warm! More advanced equipment use involves garbage bag-like vapor barriers that create a humid mircroclimate next to the skin. I have less experience with these.
Technique wise: slow down. One should never be moving so fast that one cannot hold a casual conversation. Breathing hard is an indication not of a lack of physical fitness but rather pushing to fast. (Of course, being more fit for climbing/hiking does raise the maximum speed at which one can move and chat.)
I hope this all helps rather than confuses. I'll try to answer any and all questions about this little essay on clothing and hiking. . .