Add to all of the above that the so-called "Provisional Government" which was Kerensky's only claim to fame was a self-appointed clique of notables--notables in terms of the Old Regime; none of them were involved in overthrowing the power of the Tsar and indeed the first item on their agenda was considering ways and means of putting a new Tsar in power ASAP. (They were forced to conclude this was not possible at the moment).
I'd be speaking a little bit hyperbolically to say the PG had absolutely no legitimacy whatsoever. It had a little bit of limited legitimacy--in that the non-revolutionary but still important classes in Russia supported it, by default and in desperation. But by any democratic criterion, they ought to have simply yielded to the Soviets. Note that at this point the Soviets, the self-formed "councils" (this is what "soviet" means in Russian, "council") of workers running their various factories and other workplaces and forming higher level Soviets by delegation to city level and in cooperation on a federal level, did command the allegiance of the vast majority of urban Russians. The PG got more legitimacy from being recognized as having some function by the Petrograd Soviet than it could possibly have on its own behalf. Even so, soldiers and the like regarded PG orders as real if and only if the Soviet countersigned them. Note also--at this point "Soviet" was no synonym for "Bolshevik." The Bolsheviks did not dominate the Soviets; a diversity of opinion ruled in these directly democratic bodies.
I'm not saying then that the Bolsheviks were the legitimate government. At this point they most certainly were not, not even the dominant party. I'm saying the Soviets should have been regarded as such full stop, and that insofar as the elite PG looked like an actual government, it was due in very large part to foreign recognition of it as such --and the foreign governments that really mattered were the Entente Allies. If Kerensky did anything to alienate France and Britain he would not only lose their support financially and militarily and diplomatically, he'd undercut his own claim to hold power in Russia itself! Not completely sever it as the anti-revolutionary Russian classes--the landlords, the officers, the capitalists, the middle-class supporters of these classes, etc--would surely cling to the PG and proclaim it the true Russian state--but these classes had lost control of the majority classes and their claims would be desperate and empty. Save insofar as the lower classes recognized a need for the skills and abilities of these conservative classes. But ultimately, after the Civil War, the Bolsheviks either replicated their abilities in their own ranks, or coopted former Old Regimists who concluded that the Bolsheviks had won and were therefore the legitimate government and therefore served it for patriotic reasons.
Kerensky was joined at the hip to the Entente and had no option to opt out of the war with Germany. Whether he might have preserved more power for longer by adopting a more defensive strategy is an interesting question, but the Bolsheviks were not utterly dependent on the war going badly for their bid for power, nor on German gold, though both of these things surely helped them. Perhaps if Kerensky had delayed the October Revolution by a defensive strategy, the Bolsheviks might have felt forced to offer to share power meaningfully with the radical Social Revolutionaries of the countryside, and been forced to keep them onside hence changing the nature of the post-revolutionary government. But I don't figure the self-named PG lasting long enough for Allied victory under any circumstances, or broadening its legitimacy enough to rule; eventually, with or without Bolshevik dictatorship, the Soviets could and would take all power to themselves.