Perspective:
So let's take a step back and ask what TDD is trying to help us with. TDD is trying to help us determine if our code is correct or not. And by correct, I mean "does the code meet the business requirements?" The selling point is that we know changes will be required in the future, and we want to make sure that our code remains correct after we make those changes.
I bring that perspective up because I think it's easy to get lost in the details and lose sight of what we're trying to achieve.
Principles - SAP:
While I'm not an expert in TDD, I think you're missing part of what Single Assertion Principle (SAP) is trying to teach. SAP can be restated as "test one thing at a time." But TOTAT doesn't roll off the tongue as easily as SAP does.
Testing one thing at a time means that you focus on one case; one path; one boundary condition; one error case; one whatever per test. And the driving idea behind that is you need to know what broke when the test case fails, so you can resolve the issue more quickly. If you test multiple conditions (ie. more than one thing) within a test and the test fails, then you have a lot more work on your hands. You first have to identify which of the multiple cases failed and then figure out why that case failed.
If you test one thing at a time, your search scope is a lot smaller and the defect is identified more quickly. Keep in mind that "test one thing at a time" doesn't necessarily exclude you from looking at more than one process output at a time. For example, when testing a "known good path", I may expect to see a specific, resulting value in foo as well as another value in bar and I may verify that foo != bar as part of my test. The key is to logically group the output checks based upon the case being tested.
Principles - PMP:
Likewise, I think you're missing a bit about what Private Method Principle (PMP) has to teach us. PMP encourages us to treat the system like a black box. For a given input, you should get a given output. You don't care how the black box generates the output. You only care that your outputs align with your inputs.
PMP is really good perspective for looking at the API aspects of your code. It can also help you scope what you have to test. Identify your interface points and verify that they meet the terms of their contracts. You don't need to care about how the behind-the-interface (aka private) methods do their job. You just need to verify they did what they were supposed to do.
Applied TDD (for you)
So your situation presents a bit of a wrinkle beyond an ordinary application. Your app's methods are stateful, so their output is contingent upon not only the input but also what's been previously done. I'm sure I should <insert some lecture> here about state being horrible and blah blah blah, but that really doesn't help solve your problem.
I'm going to assume you have some sort of state diagram table that shows the various potential states and what needs to be done in order to trigger a transition. If you don't, you're going to need it as it will help express the business requirements for this system.
The Tests: First, you're going to end up with a set of tests that enact state change. Ideally, you'll have tests that exercise the full range of state changes that can occur but I can see a few scenarios where you may not need to go that full extent.
Next, you need to build tests to validate the data processing. Some of those state tests will be reused when you create the data processing tests. For example, suppose you have a method Foo() that has different outputs based upon an Init and State1 states. You'll want to use your ChangeFooToState1 test as a setup step in order to test the output when "Foo() is in State1".
There's some implications behind that approach that I want to mention. Spoiler, this is where I'll infuriate the purists
First off, you have to accept that you using something as a test in one situation and a setup in another situation. On the one hand, this seems to be a direct violation of SAP. But if you logically frame ChangeFooToState1 as having two purposes then you're still meeting the spirit of what SAP is teaching us. When you need to make sure Foo() changes states, then you use ChangeFooToState1 as a test. And when need to validate "Foo()'s output when in State1" then you're using ChangeFooToState1 as a setup.
The second item is that from a practical point of view, you're not going to want fully randomized unit testing for your system. You ought to run all of the state change tests prior to running the output validation tests. SAP is kind of the guiding principle behind that ordering. To state what should be obvious - you can't use something as setup if it fails as a test.
Putting it together:
Using your state diagram, you'll generate tests to cover the transitions. Again, using your diagram, you generate tests to cover all of the input / output data processing cases driven by state.
If you follow that approach, the bloated, complicated, long, and difficult to write tests should get a little bit easier to manage. In general, they should end up smaller and they should be more concise (ie less complicated). You should notice that the tests are more decoupled or modular as well.
Now, I'm not saying the process will be completely pain free because writing good tests does take some effort. And some of them will still be difficult because you're mapping a second parameter (state) over quite a few of your cases. And as an aside, it should be a little more apparent why a stateless system is a easier to build tests for. But if you adapt this approach for your application, you should find that you are able to prove your application is working correctly.
Boring structural code doesn’t need isolated testing.
Test interesting code. That code has a behavior. Nail down the behavior you expect and not only will your code likely be correct, it’ll be easier to read.
But keep that interesting code away from the boring structural code. Do that and I’ll be able to read it and trust it without a test that isolates it.
Now if the boring structural code is part of a chain of integrated peripherals and behavior objects then fine, throw an integration test at it. If you’d like to test if I’m full of it, break the structural code and see how long it takes someone to find the problem.
Don’t waste time solving non problems. At best you’re only amusing yourself. At worst you’re actually making it harder to refactor the code.
Remember: it’s not that every function needs a test. It’s every behavior.
Best Answer
I generally write enough tests to give me confidence that my implementation is correct, but no more than that. How many tests this is depends on the problem at hand.
If you're feeling very unsure about the correct implementation of the behaviour, you'll probably end up writing a full set of tests for each endpoint and refactoring at the end: going from Approach 1 to Approach 2, as you've labelled them. Kent Beck calls this sort of programming "driving in a low gear".
If your problem is not so complex, you might want to change up a gear or two. Remember, TDD is about doing "the simplest possible thing" at every step. If you can see that the simplest way to make your next test pass is just to call the method you've already implemented, then you can go home early.
If you get an unexpected bug or test failure, you're free to slow down, write more tests, and drive in a lower gear again. TDD often feels like this: constantly adjusting your pace to match your confidence and fear levels. It's one of the themes of Kent Beck's book.
Remember, you're not aiming for 100% coverage of every public method; just enough to give you confidence that your code is correct.