Many transformers need their cords changed. That is not a reason to through it away. In my experience the windings are insulated with fish paper, various resins, and asphalt, all of which are usually in excellent condition. When I get an old transformer I usually start by checking the resistance between the prongs on the cord, assuming it is in better condition than the cord shown above. I also check the resistance to the case. If these numbers look reasonable, that is no dead short or contact to the case, I plug it into my Lionel 5D tester. This device is ideal for this as it has an switch and 4 amp fuse on the transformer outlet.
Once the cord issues are addressed I rarely find a transformer that does not work properly. Those with carbon rollers usually need them changed and everything needs to be cleaned, but otherwise they are mostly good. I always recommend the use of GFI protected outlets for old electrical appliances. On electrical appliances without a third ground prong or double insulated plastic case GFI protection just makes sense.
When opening up old transformers what I usually see is good solid engineering appropriate for the period and high quality handwork craftsmanship. I have spent much of my life working on old electrical equipment. When working on this equipment I often think about the people who made it. They were not watching automated assembly line fabrication, they were at a bench with hand tools, rudimentary machines and the necessary materials making this equipment. Hundreds of hardworking people, many of them woman, working long days for not much pay. Here we are 100 years later, long after they are dead, still using equipment they made.