Yikes! Don't plug those transformers in. That wiring is an accident waiting to happen. As to your question of why you locomotive is not working, people far more knowledgeable than I about pre and post war Lionel will answer your question.
Definitely have those transformers rewired and serviced at a service center if you are not up to doing it yourself.
The track can be cleaned up with EvapoRust if you do not mind handling caustic chemicals and getting dirty; if not replacement may be better.
The uncoupling track can be cleaned up and rewired with 4 conductor ribbon cable.
As for the locomotive, it probably needs a total overhaul of the motor and 'E' unit, with a good lubricating to get it running. You can do it with the right tools and knowledge; otherwise leave it for your local serviceman.
All of that stuff is in really bad condition. Be careful and don't expect anything to work. right now so much of it looks old and deteriorated that there would be n hope of figuring out what was wrong. Very likely, almost everthing is not working as it should.
For the very basics, you need to first look at videos that have been posted here by Benz Trainz both here and on youtube. Benjamin has a good way to clean track and remove rust. He also takes inoperable locomotives and makes them work again. After you watch about a dozen or so of his videos, you may want to embark on attempting to get everything up to being operable. Or, you may just opt for purchasing some trains that are currently in running condition.
Either way, you will gain some great insight and knowledge from Ben. Simple fixes at little or no cost, and they will be fun to do.
Since your original post was to seek knowledge, I know of no other place than with Ben's videos. Start there.
Bens videos are probably easier to find on YouTube then here. Here is a link to one for derusting track using a solution of white vinegar 5% solution.
Good luck, I think you will be pleased once you start watching Ben at work.
The 2026 is a good starter engine. A top screw and a rear rod pushed through the shell will remove the motor. The rod will only push one way. I can't remember which maybe left if the shell is upside down. One end is larger than the other.
BY removing two screw you get to the armature. take a picture if you can and clean the area.
The AF transformer may be the best choice if the cord is good.
Everything is in rough shape but not impossible.
How clean is the motor and reverse unit?
Out of storage I have gotten an engine like this tested in about twenty minutes.
as everyone else says.. be very careful if the transformer wires look brittle and bare else you will electrocute yourself.
IF they are ok enough or open it up and replace cord yourself... I just did this on my ZW and used a replacement lamp cord for $10...
then turn on transformer and give it throttle.. see if there is voltage on the output pins that would connect to the track. need a volt meter of course.
for the track vinegar post above, I did this to A TON of track. over 100 pieces and it worked great. I had to let some sit for 4-6 days!!!!! probably gonna let them sit back in it again.... good but not perfect.
then if clean track and voltage from transformer, connect the two together without a train... measure voltage on the track at variable spots.. and see if you have juice...