If I am offered the choice between tea and coffee, do I have 'libertarian free will' or is my free will overturned by the fact that I prefer coffee to tea?
If men are offered the choice between sin and righeousness, do they have 'libertarian free will' or is their free will overturned by the fact that they prefer sin?
John 3:19. 'And this is the condemnation, that the light has come into the world, and men preferred darkness rather than the light because their deeds were evil.'
What is needed is a change in men and women so radical that is can only be described as a new birth - a birth of water and the Spirit (John 3:5). 'I will sprinkle clean water on you, and you shall be clean: I will cleanse you from all your filthiness and from all your idols. I will give you a new heart and put a new spirit within you; I will take the heart of stone out of your flesh and give you a heart of flesh. I will put My Spirit within you and cause you to walk in My statutes, and you will keep My judgments and do them' (Ezekiel 36:25-27).
They do not have libertarian free will. Even among atheists there is a dichotomy between desires and a choice must be made.
A choice between tea and coffee? I would probably choose coffee as I like coffee better than tea. But on occasions I do choose tea. Obviously out desires affect the choices we make.
That is why I say that Arminianism does not even come close to suggesting libertarian free will.
Generally we are talking about salvation (not coffee or tea). So let's look at it.
Arminianism (and most free-will theologies) hold that natural man will only choose to reject God because they are "flesh". But the Spirit works within man in such a way as to influence their desires (by the Spirit men can choose God).
That said, man has these two competing influences (the flesh and the Spirit). Man freely chooses one of these two paths.
That is not libertarian free will as both choices are influenced choices freely chosen. Free-will is the ability to choose between two opposing options.