I came across similar findings. Here's just one example,
The verbs could, should, would, might, must, needn’t, ought to, used to do not normally change.
Example: He said, “She might be right.” – He said that she might be right.
The very same questions you're asking are being asked and debated here and here by other students. You may want to take a look. If you have questions, let us know.
All the best.
How Do We Overcome the Difficulties of Teaching Conditionals?