You say, “Assuming logic and rationality is necessary for any kind of argument or discussion.” Forgive the tautology here, but assuming logic and rationality is necessary for any logical or rational discussion. There are other kinds of discussions – emotional appeals, for example. But, limiting ourselves to only logical, rational discussion, as scientific communities aspire to do, then your statement is correct, and in fact that is one of the main points of TAG! If we want to engage in rational debate, then let’s start by assuming that logic and rationality exist and see what that entails, necessitates, or requires. TAG argues that some basic notion of God is required. (In this way, it follows the format of other transcendental arguments in philosophy, like those of Immanuel Kant.) The main thrust of TAG is to show that, on assuming premises that most people do already, God is already there!
In hindsight, then, on what foundation can we make the logic and rationality assumptions, ones which we take for granted on a daily basis? The intellectual tools that we can’t help but use lead us to God. Consider the contrapositive of TAG: if God does not exist, then the argument would argue that the universe could not exhibit the standards of logic and rationality which are so foundational to our daily lives. For a Christian, this should not feel like an overly ambitious statement, for we know that all that we have flows from God and His provision.
Now, on to a few other points. Your comment has helped me realize a few things: first, the use of the word “universal” needs some more clarity. Here, I intend universality (in morality) to mean that the moral principle applies to all people, not that all people agree that the principle is a moral. It is universal in its application and authority, not in its agreement. In fact, none of these 3 is universal in the latter sense. There is not universal agreement about logic (for example, some hold to the “law of the excluded middle,” and some do not), or rationality (for example, different philosophical schools which disagree about what constitutes proper reason). Universality in agreement is not what I am after, but universality in application. Thank you for pointing out this vagueness.
Secondly, on the personal nature of morality, what TAG needs is just the existence of at least one moral standard which is personal (and the other characteristics mentioned). For the purposes of TAG, morality which is not inherently personal could exist. (I happen to believe that all morality is ultimately personal because anything morally wrong offends our personal God, but of course this is not part of the argument – it is a “hindsight view” which would be a blatant circularity). So, if you agree that at least one personal moral exists, then that premise of TAG is met. Again, I thank you for identifying this area of improvement in the argument.
Why is morality included as a premise? I have a few thoughts here. First, the “formal” one. The structure of TAG is a logical argument, so it could include anything at all as a premise. The discussion would be whether or not the premises are accepted, but there need not be justification for why they are chosen. The point is what those premises imply in the conclusions. If the premises are accepted and the argument’s reasoning is logically valid, then the conclusion is unavoidable, regardless of why the premises were chosen initially (if there was a reason at all!).
But seeing as how you would like an intuitive reason for morality’s inclusion…morality is one of the authoritative standards that people inevitably appeal to. Implicit in so many discussions is a notion of what someone ought to do or think or say (or ought not to do or think or say). A notion of “moral ought” seems to be baked into humanity. Logic, rationality, and morality together are the “big 3,” in my opinion. Thus, these 3 are chosen because they are foundational to our everyday experience. But I’ll reiterate that the real question is what those premises, if accepted, imply: TAG claims that it is the existence of (some kind of) God.
And finally, if you find apologetics destructive of your faith, then I think you can ignore them. What I want most is for each of us to have saving faith in Christ, and I recognize that the Holy Spirit can accomplish that in many different ways in a person’s heart. Apologetics have been very useful to me and many others, but I won’t twist anyone’s arm to care about apologetics if they are counterproductive to the real goal: faith in Christ in that person’s heart.