The problem with neural nets is that the inputs (the game situation) has to be fed to it as a bunch of numbers.
That means that there usually is a heck of alot of interpretation pre-processing required to generate this data.first.
You can feed images to a CNN these days.
Another problem is that the process of 'training' the neural nets is usually only understood from the outside - the logic is NOT directly accessible to the programmer. Alot of 'test' game situational data needs to be build up and maintained, and connected with a CORRECT action (probably done by a human) to force the neural net into producing is required. Again alot of indirect work.
You can make the network return an estimate of future rewards for each possible action: Read the DQN paper I linked to earlier. There are mechanisms to look into what the neural network is doing, although I think it's best to use NNs in situations where you don't particularly care how it's doing it.
Neutral nets also generally dont handle complex situations very well, too many factors interfere with the internal learning patterns/processes, usually requiring multiple simpler neural nets to be built to handle different strategies/tactics/solutions.
That's not my experience.
Usually with games (and their limited AI processing budgets), after you have already done the interpretive preprocessing, it usually just takes simple hand written logic to use that data -- and that logic CAN be directly tweaked to get the desired results.
That is the traditional approach, yes: You define a bunch of "features" that capture important aspects of the situation, and then write simple logic to combine them. When you do things the NN way, you let the network learn the features and how they interact.
It might be that practical neural nets may be just a 'tool' the main logic can use for certain analysis (and not for many others) .
I think you should give NNs an honest try. In the last few years there has been a lot of progress and most of your objections don't apply.
If you can define a reward scheme by which the quality of an agent's behavior is evaluated, you can probably use unsupervised learning to train a NN to do the job. I don't know if this is practical yet, but with the right tools, this could be a very neat way of writing game AI.