If Kornilov struck first, surprising unprepared Bolsheviks, and may I remind everyone a lot of them were in prison at the time until Kerensky released them IOTL, then how can they launch any revolution if they got worse than decimated?
Just to let you know, Shliapnikov, the Bolshevik most involved in organising the party in Petrograd, had huge support amongst the workers there and was vital in the organisation of the Red Guard
wasn't arrested at this point. And neither were thousands of Bolsheviks despite Kerensky's efforts including Lenin who was in hiding.
OTL October revolution was really an armed coup, not popular uprising. Rank and file soldiers might not supported whites, but neither did they supported reds.
During the Moscow Municipal Duma elections prior to the October Revolution, out of the 17,000 soldiers who voted in that election 14,000 voted Bolshevik. Conscripted soldiers were specifically supportive of the Bolsheviks because the Bolsheviks were the only organisation calling for the end of the war.
With support of Kornilov, Kerensky would maintain control of army (which was why Kerensky made Kornilov Commander-in-Chief in the first place, to placate military), and any counter-attack by Bolsheviks/Mensheviks/else would fail. Because whites would got rid of large number of their enemies, and maintained control of capital city.
No Bolsheviks, no Bolshevik revolution.
During the July Days there were roughly 10,000 workers organised in Red Guards in Petrograd and they weren't specifically Bolsheviks - they were organised by factories and the soviets. By September/October there were roughly 20,000 Red Guard and 79 factories in Petrograd were organising weapons training for their workers. Technically these Red Guards were under the command of the soviets themselves and they had Bolsheviks, anarchists, SRs and others as commanders.
After the July Days, Kerensky organised commissars to board ships and arrest agitators and they received the reply: "We are all agitators." The Kronstadt Soviet, with its thousands of armed sailors barracked near Petrograd, had refused to recognise the Provincial Government since May. Bukharin relates to an instance of supposedly loyal soldiers guarding the Duma asking the Bolshevik delegates for when they get to use their rifles. Frankly put: the rank and file soldiers, conscripted into a war they didn't want to fight, didn't support the provincial government and would defend the soviets if Kornilov and Kerensky moved to have them purged.
Keresnky was unpopular amongst the workers, particularly in Petrograd, but he had little support even within his own party. In June during a congress of the Social Revolutionary Party he was voted down from the party's central committee, receiving only 135 votes out of 270. He was basically a figurehead of the establishment, of the bourgeoisie, but had little support amongst the workers and soldiers who would have resisted any attempt to purge the soviets with a general strike at the very least but more likely armed insurrection.
As for Kornilov? Miliukov, the Kadet leader, writes his depiction of the State Conference in Moscow when Kornilov got up to speak: "The short, stumpy but strong figure of a man with Kalmuck features, appeared up the stage, darting sharp piercing glances from his small black eyes in which there was a vicious glint. The hall rocked with applause. All leapt to their feet with the exception of ... the soldiers." By all accounts, from the writings of General Brusilov, Miliukov and others, Kornilov was a poor organiser who didn't know much about politics and lumped all those participating in the soviets together. He would have just as likely tried to hang Chernov or Martov as he would have a Bolshevik. The soviets in their entirety would have resisted him in Petrograd. Insurrection was inevitable: hence Kerensky trying to play both sides in OTL.