While it's interesting that the Russian leadership would make an explicit statement like this when they have the opportunity to leverage a degree of strategic ambiguity to prey upon the hopes of the Trump admin's Kissinger-wannabes trying to explicitly manifest such a thing - in my cynical interpretation, this signals a certain anxiety within the Sino-Russian relationship that such a thing might actually be possible, which Lavrov is thereby compelled to go out and publicly reassure - the issue is Russia's historical track record.
Under Gorbachev, it threw all of socialist Europe under the bus. I've been reading Honecker's prison memoirs lately and he personally attested how the DDR government was not even given a place at the table in the negotiations for German "reunification." Gorbachev went over a Soviet ally of half a century because he was so desperate for the likes of Reagan, Thatcher and Kohl to shake his hand and chose to negotiate with them directly in the same way, ironically, that the Trump admin is sidelining the Ukrainian regime in recent weeks. A statement like this could very well have the same lack of worth as Brezhnev's saying that the "USSR would never betray its Warsaw Pact allies." It's a frankly trite truism, but also one that already rang true for Russia in the past, which is that no one can predict the personality of future leadership.
The real impediment in the attempts to foment a Sino-Russian split 2.0 is the American desire to have its cake and eat it too by refusing to give the Russians any long-term geopolitical concessions. What the current Russian government led by Putin has always wanted is a "G2" with the US over Europe. This naturally goes against the American objectives in Europe, dating back to the Cold War NATO founding principles of "keeping the Russians out." A "G2" relationship requires concessions and strategic sacrifices that the US is fundamentally and psychologically unwilling to make. This is what scuppered the Obama era "G2" proposals with China, which basically ordered China to remain in place as a permanent toy and clothing factory while denying the Chinese even the concession of reclaiming Taiwan.
The pathway to "split" the Sino-Russian relationship is actually there, in my view, so long as the following conditions are achieved:
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Allowing Putin the room to attempt to reclaim a sphere of influence over Eastern Europe in the same way Churchill in his memoirs alleged he brokered with Stalin the post-war European orientation on napkin paper.
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Forcibly ordering the economic binding of NATO Europe to Russia and solidifying this dependency to wean Russia off the material conditions of its partnership with China, which is primarily the immensely expanded economic ties.
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Forcing through bi-partisan domestic American allegiance to this "G2" alignment by enshrining it as treaty or legislation, so that it can't be repudiated by a later presidency, as this would be the only way to ensure Russian trust to any such arrangement.
I'd say this latter point is actually easier done than might be believed: the anti-China agenda is the leading bipartisan consensus in Washington and a coordinated narrative campaign framing such a move as "owning China" and blasting any political opposition as "surrendering to China" would overcome even the Democrats' Russophobia through their even more fanatical Sinophobia.
The outcome, however, would not be a Russo-American grand Christian white alliance against "the East," that Nicholas II dream of the Republicans, but a geopolitical paradigm where Russia is incentivized to openly hedge against China and the US as a much more "neutral" party.
The issue for the US is that it is fundamentally unwilling to pay the high price for such an arrangement. Going from having defeated the USSR, shattering it into partitioned pieces and successfully making those pieces fight against each other for US interests to being forced to share Europe with the territorially diminished Russian successor state is too much dramatic a psychological transition for Washington's foreign policy blob to contemplate. This Western chauvinism of being unable to accept a relationship of true equality with a non-Western state, with all the implications and sacrifices that entails, is the colossal impediment to the "Reverse Kissinger" aspiration of "getting Russia to betray China."