At the time of 'I, Borg', the federation and the borg were in a state of war (since 'best of both worlds').
In war, in the vast majority of cases, one doesn't have a morally 'white' option.
One only has 'grey' options - either you kill the enemy, or he kills you. One can only choose the 'lesser evil'.
About the borg - in 'Scorpion', Picards' words established that ~'One should not expect reason or mercy from the borg. In their collective state, they feel no pity, no remorse, only the will to conquer
'. The episode also established that the borg have at least hundreds of thousands of ships.
The borg attacking and destroying the federation is only a matter of time. They can't be reasoned with, they can't be defeated - normally. An implacable, unstoppable enemy.
And yet, even in this situation, Picard was still unwilling to accept that he has no morally correct, 'white' option.
So - that choice did he make? The choice that, on the short term, was apparently moral (not using Hugh), but in the medium - long term, was 'pitch black', by far the 'greater evil'.
Since 'I, Borg' until VOY's end, how many species did the borg assimilate? How many BILLIONS
did they kill? How many BILLIONS
did they assimilate? Death and suffering beyond human comprehension.
The blood of all those people stains Picard's hands, too, because he could have stopped it all and he didn't.
As for the invasive 'logic paradox' not working - Data/Geordi were sure it would work - and given that their knowledge of the hows and whys of trek tech (even borg tech) vastly surpasses what we posters know (for example, they, too, know the borg adapt and, unlike us, they know much about how the borg do this magic trick), the chances are this 'logic paradox' would have dismantled the borg hive mind.