But they have allowed divorce followed by remarriage (adultery) since the schism, and officially allow birth control since the 1970s.
They might have less (open) Reinhard Marx's or De Kesel's but they also seem to have zero Vigano's, Burke's, or Eijk's.
We have Pope Benedict XVI. They Kirill.

Yes, but Kirill didn't turn coward and quit, Benedict XVI did. When in office, Benedict XVI was a source of great pride for Catholics, especially with Summorum Pontificum. Now, for millions he is a source of great disappointment, and even shame. Had he not turned and ran from his office/responsibility, we would not be in the disaster we are today. Benedict XVI, either knowingly or not, opened the door to a heretic, and his associates who are destroying the Catholic Faith.
Remember the thunderbolt that hit St.Peter's twice the night Benedict XVI quit? THat wasn't God showing his anger....that was the door opening to Satan. It was a perfect opportunity for Him and his legions of Hell....and he has made good use of the nearly 10 years since.. So don't say "We have Benedict XVI, They have Kirill" as if degrading Kirill.
We have nothing to be proud of for Benedict XVI anymore.

Kirill hasn't been faced with any neo-marxist conspiracy. In fact, Kirill worked for the KGB. Kirill blindly supports Putin and neo-Sovietism. He is alligned with the princes of his nation. He has it easy. Russian orthosoxy in general had it easy. They reached an accord with Stalin, were happy to accept the churches Stalin stole from the Eastern Catholic.
The Communists didn't find it necessary to sent thousands of marxist pederasts into the seminaries of the schismatics. Their target was an is Catholicism.
We don't know whether Benedict XVI turned a coward. He quite the ministry but whether that was because of mere cowardice or things he knew regarding (end times?) conspiracies that we don't, we cannot be sure of yet.
Saint Peter denied our Lord 3 times.
Question is what more Benedict XVI was capable of doing in 2013.
Kirill hasn't tried to expose and destroy the pederasts amongst the Eastern schismatics. The Russian media speaks little off it. Same in Serbia. One of the few priests who spoke out regarding abuse in Eastern orthodoxy got defrocked.
How is Kirill better than Benedict XVI? He is worse. Benedict XVI may have retreated from the fight. Kirill doesn't fight for Christ or his Church but for the Russian state religion and the Russian regime. He barely has to fight. He is the spokesman for the fighters.

I sympathize somewhat for Ukraine, but it was always part of Russia. They only broke away after the Soviet Union fell apart. NATO and especially USA should not get involved with Ukraine, or Putin. Putin doesn't want to restore the Soviet Union, instead, he thinks of the greater Russia that existed under the Czarist regimes. He even wants to restore the Czar, but in a figurehead role like King Charles III with Putin actually in control (or his Prime Minister successors).
Kirill supports Putin because they all have an idea of "Mother Russia". or in Czarist times "Holy Russia". Very Nationalistic. So Kirill would never go against Putin.
zBenedict XVI WAS a coward to quit. It had nothing to do with his health (he's still living at 95), or the "end times". It had to do with the MASSIVE homosexual scandal of the Vatican, which has been allowed free reign under heretic Francis.
Benedict XVI didn't know how to correct/cope with it, so he quit.

(Parts) of Ukraine only came under Russian rule in 1686. The Cossack Hetmanate fell under Russia rule and Habsburg rule later, similar to how Poland fell under Russia and Prussia.
While I don't believe Putin is a Communist and consider your claim that he supports a more classical great Russian nationalism very plaussible, he did restore the Soviet anthem in 2001.
Worse, he has outlawed historially accurate statements regarding Molotov-Ribbentrop. Defended Soviet horrors against Poland.
I also find it telling that with a supermajority in both houses, Putin didn't pass a constitutional amendment protecting unborn life. The gay marriage ban was lowhanging fruit, but it doesn't seem like Putin is willing to crack down on more popular sins. Nor does the patriarch seem to pressure him in any way.
The Catholic Church in Poland has done far more even while Soros money was sent to create mass protests.
Benedict XVI was cracking down on those homosexual scandals and what caused him to retreat exactly, I don't know.
At least his loyal prelates like Vigano, Muller, Burke, Eijk, Sarah, Zen, Pell and so on are bravely continuing the fight.
Pell and Zen are true examples of martyrs.
But Kirill has not shown any interest in antagonising the pederasts within orthodoxy. He seems to be fine with it being swept under the rug.
Fr. Kuraev got defrocked for speaking up about how at least 14 % of the orthodox bishops had covered up abuse.