I think utabintarbo's axioms are just assumptions too. They have justifications, but then, so do other axioms. "Existence exists" follows from assuming ones own existence. Existence of consciousnesses because denying it would deny your own consciousness is not a proof either. (argued in #132) I do not follow "everything that exists its own nature"(Law of identity) reasoning either. I would say that the Law of identity is not really needed to base ethical axioms on, for that you assume that you can get a good guess on what responses of things are, in such a way that one can affect things in a way improving the chances of reaching a goal.

You are talking about Ayn Rand Importance Of Philosophy-site utabintarbo links to in his site. (that link is broken for me btw) I do not think he just regurgitating it, but as anyone, it is probably a good idea to critic ones own beliefs.(not saying you dont)
I have read a large chunk of that website, disagree to it very much. For one, being fed is not a human right according to the website, is depriving someone of food not 'force'. I mean, that person may not have ways to reasonably trade for food, given that, that person will be forced to do something. Either being deprived of freedom by having to work all his life for food, or by initiating 'force' against those who are in (much)better state.(or even only the latter) (here 'force' refers to how it is used in the website)

Much of what that website is in the 'ethical axioms' category. (Although i cannot say if it is axiomatically set up at this point) The ethical axiom seems to be self-interest, one owns life. The website claims (erroneously in my point of view) that self-interest is at least usually beneficient for the rest of the group, and also never includes initiation of force.
I agree on with the website, that the assumption of an 'absolute good' is not good enough, it does not specify at all what this good is, this absolute value is not determinable. Feel-good does specify it more, but one could still take pain and misery as feel-good, but that would mean one would like pain and misery oneself.(or that person is unreasonable?) Feel-good defined with observations of what consciousnesses do when given freedom(define as open options) has complications with people that have mental issues, but is better defined. Then there is the problem of what the weighs should be.
Admittedly, self-interest is one ethics-axiom that does have few issues with defining it self, as it is do what you want. Perhaps it can come very natural, survival of the fittest and all. But as the website says, humans have survived because of our brains, so i think humans should keep surviving by design, not by evolution.

I would like to add, the consequences of ethical axioms strongly depend on the nature of consciousnesses and things. Assuming feel-good axiom, if i was a stone(d) ager, i might think that everyone should judge at the spot, by themselves, from their own knowledge, the best thing to do in circumstances. Indeed, in that case that might even be true, but i much doubt that it is true in our age. In our age i might thing of free speech and promotion thereof, law (enforcement) as things to help us in our goal(s).