You're not wrong, of course.
The difference between then and now is to do with scale. The changes that you are talking about happened over the period of many hundreds of thousands/millions of years until a runaway effect took place. What we're seeing today is many thousands of years of climate change happen within about 50 years or so, which is extremely concerning.
The Earth's climate seems to be a very finely balanced thing. The introduction of only a small bit more carbon dioxide molecules into the atmosphere has started the slant. Human's only actually produce a very small amount of the overall CO2 in the atmosphere but it's enough to start the instability. Naturally produced CO2 is absorbed by various things on the planet ("carbon sinks") so that the equilibrium remains. An analogy for this is a glass of water. Let's say you have a 1 pint glass of water. Every hour you drink half a pint and every hour the glass gains half a pint of water. You're in a perfect balance and the glass never runs dry or overflows. Now let's say that instead of giving you half a pint of water every hour, you now get half a pint + 1ml. 1 millilitre of water is nothing compared to the pints and pints that are been poured into the glass every single day but given enough time, that 1ml sends the whole system out of whack and the glass overflows.
That's what's happening on Earth. Climate change is a globally studied issue, across a ton of seperate fields. It's not just climatology where we get our evidence from. There's evidence from physics, from astrophysics, from geology, from biologists or conservationists, etc. It is possibly now the best studied and most well funded scientific model in the history of the world - mainly because it has the potential to be the most damaging, in part because it's an extremely complicated subject that requires linking together thousands of little sub-systems into a big picture, and also in part of course because of the politics of the issue.
But with that funding has come rigorous debate, skepticism and discussion. I'd argue that climate change is also one of the most challenged scientific models that are currently in wide circulation too - and that's a good thing. Science needs constant challenging of models set in place in order to prove the correctness of its ideas. And climate change, for all of the challenge that has come from legitimate skeptics inside the scientific world who have actually sensible objections to parts of it, has still passed all of these and seems to explain the world around us better than any other model.
With science that's all we can do. We look at the world; experiment, observe and predict. When we come up with an idea that fits observed data, and most importantly can predict future data accurately then we adopt this idea until something that fits better comes along. Despite many attempts by many different groups, nothing has fit observational, experimental and predictive data better than the idea of anthropogical climate change. I wish it would. I'm pretty sure everybody does. The idea that actually we're not heading for a global crisis which will destroy whole economies and create millions of refugees while taxing the world's food and water supplies above breaking point is comfortable and is certain preferable. But we can't ignore what we see in the data, no matter how uncomfortable it is, we have to be intellectually honest.
