"It is especially cynical that legally insignificant decisions ... were made public on the eve of the 20th anniversary of the US illegal invasion of Iraq, where the ICC had jurisdiction, but did nothing to bring the perpetrators to responsibility," Nebenzya said.
That's one public statement from a Russian official, in the article you cited in the question, expressing cynicism on why the Russians don't have much faith in the ICC.
A major reason why it is perceived to be a "western" institution is because it is not part of the United Nations - European nations are its most staunch advocates and supporters. In fact, it is an EU requirement for a member state to have ratified the "Rome Statute" - meaning signing to become a member of ICC and accepting its jurisdiction by passing a national legislation:
All Member States have ratified the Rome Statute. ...
The principles of the Rome Statute, as well as those governing the functioning of the International Criminal Court (hereinafter the ‘ICC’), are fully in line with the principles and objectives of the Union ... The Union is convinced that universal accession to the Rome Statute is essential for the full effectiveness of the ICC ... Council Decision 2011/168/CFSP
African states too were vocal advocates for the ICC and played a big role in its creation. But after the establishment of the court, it has become quite cynical of it claiming that only African black leaders are being specifically targeted by the ICC (which they also accuse of being a "European court"):
Increasingly, more and more African nations are expressing their dissatisfaction with the operations of the court... criticism directed at the ICC by Africans is that the court has tended to focus almost exclusively on Africans and — more specifically — black Africans...
A common criticism of the ICC by Africans is that the court is subject to political manipulation... Recent data show that about 60 percent of ICC funding comes from the European Union. Like other international organizations, it is claimed that there is a close relationship between funding and influence over the activities undertaken by the court.
A related concern is that the West also holds influence in that most of the substantive appointments — the key positions that are responsible for investigations and the preparation of cases — are overwhelmingly staffed by members of the European Union. For the ICC, there is a tight relationship between funding and staffing. This close relationship has raised justified concerns that the court is a tool of the EU, which, unfortunately, reinforces the “new colonialism” sentiments. - Can the International Criminal Court play fair in Africa?
In the middle-east, Palestinians too have become disillusioned and frustrated with the ICC (Palestine is a member state of ICC):
[ICC prosecutor] Khan’s office has “an ongoing investigation with jurisdiction over Palestine that goes back to 2014”, he stressed. One cannot help but ask how the ICC managed to find Russia guilty of war crimes in Ukraine and issue an arrest warrant for President Vladimir Putin within a year - yet after nine years, there seems to be no urgency to complete the investigation of Israel’s recurrent war crimes and bring the perpetrators to trial.
Ironically, the concerns voiced by the Russian are actually similar to what the United States also once publicly said:
Although initially a supporter of the proposed Court, the Clinton Administration did not sign the treaty at the Rome conference because of a variety of concerns, including a concern that the treaty contained insufficient protection against politicized prosecutions.
("Politicized prosecution" is precisely what the Russians feel is happening today against Russia / Putin. A good analysis on the legal objections that the US had against the ICC is here - The International Criminal Court's antagonism toward our Constitution).
Despite the US hostility towards ICC holding a trial for any Israeli or American for war crimes or genocide, the US hasn't had any problem in using the ICC for their own political needs:
In 2005, the Bush administration allowed the Security Council to refer the Darfur case, and it later offered to assist the court’s investigation ... In 2011, the Barack Obama administration voted in favour of the Security Council referral for a Libya investigation. It also helped deliver several fugitives to The Hague and offered to pay millions of dollars as rewards for information on individuals accused of atrocities. The Role of the International Criminal Court
But when the ICC wanted to try US personnels for crimes in Afghanistan, or Israel for their activity in Palestine, President Trump even tried to sanction ICC.
Now, in the context of the Russian - Ukraine war, and Biden's proclamation that Putin is a war criminal, some US politicians have even introduced a resolution in the US Senate advocating for Putin (and other Russian officials) to be investigated and tried in the ICC:
Resolved, That the Senate—
(1) strongly condemns the ongoing violence, war crimes, crimes against humanity ...
(2) encourages member states to petition the ICC and the ICJ to authorize any and all pending investigations into war crimes and crimes against humanity committed by the Russian Armed Forces and their proxies and President Putin's military commanders, at the direction of President Vladimir Putin;
(3) supports any investigation into war crimes, crimes against humanity, and systematic human rights abuses levied by President Vladimir Putin, the Russian Security Council, the Russian Armed Forces and their proxies, and President Putin's military commanders...
In this background of EU backing of the ICC, the criticism of the African Union of ICC being "European" and biased against them, the disillusion of the Palestinians and hypocrisy of the US advocating for Russians to be tried in ICC despite their own antagonism towards the ICC, it should be easy to understand why the Russians view it with such cynicism and call it a "puppet of the west".
Also, the claim that the ICC is neutral because of the number of judges of different nationalities is a bit politically naive when each of these judges can (and will) act on their national interest - they are, after all, political appointees of their nation-state. Thus, there is a real possibility that the judges can be politically influenced through their country. This is important as only two-thirds majority is required for conviction in the ICC.