I've said this before, but there were legitimate grounds for Russia ending the war with Prussia in 1762. It wasn't just a case of Pyotr being an unconditional, fawning admirer of Friedrich's (though he was). There was a wide base of support in the Russian aristocracy for ending the Prussian adventure.
Apart from Elizaveta's personal hatred of Friedrich - the overwhelming reason Russia entered the war in the first place - Russia had a few foreign policy goals in entering the war. The first was that it was part and parcel of the Austrian alliance that Russia had maintained for several decades. By 1762 this was increasingly a dead letter. Russian commanders considered the Austrian army to be incompetently staffed and led, while the Austrians believed that Russia's sense of priorities was wrong and that Russian armies moved so slowly as to indicate that they did not want to fight at all. The course of the war had caused considerable friction between the Habsburg and Romanov courts and on more than one occasion it was Elizaveta's own force of personality that kept cooperation going as long as it did.
A second was that Russia thought that it could potentially gain East Prussia as a territorial prize. This was a little on the ridiculous side, but it had some merit (especially since the Russian occupation forces and administration in East Prussia found that the Junkers there were more than willing to cooperate with them). By 1762, however, Russia had found that neither the French nor the Austrians would be willing to agree to Russian annexationist demands, for various reasons, and Russia did not yet possess the military and diplomatic power to be able to do whatever it wanted regardless of the consent of other powers. The territorial demands, probably unrealistic from the start, were therefore virtually impossible to obtain.
Russia had also gone to war to check Prussian influence in Poland-Lithuania, and this war aim assumed a larger dimension in 1758 and 1759 as fighting shifted eastwards into the Wielkopolska and Neumark. But after the battles of Zorndorf and Kunersdorf, Prussia lacked the military power or the political influence to seriously contest Russian domination even in western Poland. This goal had, therefore, already been accomplished as well as it could possibly have been.
Added to all of these was an increasing sense of the cost of fighting. Russia had lost a lot of men during the fighting, especially at Zorndorf, Kunersdorf, and in the siege of Kolberg. Fighting on the other side of Poland-Lithuania was also not cheap, either in terms of cash or in kind, and also involved considerable troop wastage. While the possession of East Prussia defrayed the costs, increasingly the Russians had to rely on Austrian commissary support and French money to maintain their army in Central Europe, and both of these things were becoming increasingly scarce as the Bourbons attempted to wind down their military effort and as Habsburg finances began to break down.
Finally, it was very clear that, even in 1762, Prussia retained considerable powers of resistance. Even though Friedrich had been defeated by most reasonable measures, he retained sizable armies in the field and Freikorps throughout his domains. The battles of Liegnitz and Torgau, the campaigning around Bunzelwitz, and the fighting between Heinrich and Daun in Saxony demonstrated that Prussia was extremely dangerous even in this straitened state. Forcing Friedrich to recognize the conquest of East Prussia, the only theoretical remaining Russian war aim, had already incurred costs out of all proportion to the potential benefits, and would certainly be even costlier going forward.
So in a very real sense, Elizaveta's death was not an unnatural and unlikely act of salvation for the Hohenzollern king. It was her obstinate refusal to stop prosecuting the Prussian war that was "unnatural". Of course, Pyotr's decision to switch sides and permit Friedrich the use of a Russian auxiliary corps was ridiculous. If there was little to no benefit to Russia in fighting Friedrich, there was even less benefit to fighting Maria Theresa.