If they take their computer model, and plug in data from, say, 1900, will it accurately show climate data from 1950? If not, why not?
Indeed. I seriously contest the accuracy of their model. If it is truly accurate, do as you say. And further it should be able to accurately predict every major hurricane over the last century.
If they use the measured data after the "correction factors" are incorporated then the old data will match the computer model. If you use today's measured data the computer model will not match future measurements because they don't know what "correction factor" to use on today's measurements yet. It is a not-so-clever hoax, but people fall for it.
The easiest thing in the world to model is history. You can always take a set of data and use them to fit observations. The tricky part is getting your model to reliably predict the future. Now, from what I have read in the popular literature, climate models do not do a good job of predicting (postdicting?) the past.
There are useful and well established ways to rate a model, to tell if you are "overfitting" the data, and to know how well you can predict the future assuming you are observing a stationary process, meaning the the system you are modeling does not change, only the inputs (e.g., CO2 levels) change, and that you have account for all the important variables (e.g., solar flux).
Again, from the little I can glean from the popular press, there are no reliable models that have had any success in predicting climatic changes, and the only controlling variable used is CO2. As Roy Spence of the UAH has pointed out, we cannot rule out the possibility that the sensitivity of atmospheric temperature to CO2 is negative. Meaning that more CO2 means a (slightly) cooler not warmer planet.