If God hardens a person's heart, he does not "allow" the person to believe. Thus to allow or permit is clearly the contextual meaning of the word in John 6:65.
I would point out that the hardening that takes place can probably be attributed to God's efforts. Consider Pharaoh: it was God's commands that resulted in the hardened heart. So His efforts come before the hardening in that case. This is a reasonable conclusion in general, because one cannot reject the Gospel prior to God's enlightenment.
Stephen makes a general statement concerning the "fathers," that they always resisted the Holy Ghost:
Acts 7:51
King James Version
51 Ye stiffnecked and uncircumcised in heart and ears, ye do always resist the Holy Ghost: as your fathers did, so do ye.
It is after He has sought to enlighten men that He brings about conditions where they cannot believe. An example would be,
2 Thessalonians 2:10-12
King James Version
10 And with all deceivableness of unrighteousness in them that perish; because they received not the love of the truth, that they might be saved.
11 And for this cause God shall send them strong delusion, that they should believe a lie:
12 That they all might be damned who believed not the truth, but had pleasure in unrighteousness.
Before placing too much emphasis on "believed not," take note they did not receive the love of the truth. That is why they were not saved, rather than not believing. Being unbelievers is the inevitable conclusion for those who receive not the love of the truth, or, as Peter puts it ...
2 Peter 2:21
For it had been better for them not to have known the way of righteousness, than, after they have known it, to turn from the holy commandment delivered unto them.
There is no hardening of hearts prior to God's ministry. There is no faith, no belief, no repentance.
God bless.