do they think Russia is going to do any less damage to the world than America if they get the chance

Very probably, yes.

Around 20 years ago Russia—at the time lead by Putin—wanted to join the imperialism club, but the US rejected them. Ex-Nato head says Putin wanted to join alliance early on in his rule. Since then Russia, rejected by the Global North, has had no choice but to join with the Global South as allies instead of neocolonizers. Hence BRICS+ and the larger developing multipolar bloc that’s going its own way, ignoring the US’ “rules-based international order” sanctions, developing its own international balance of payments outside of US dollar hegemony, and working to get out from under the boot of the IMF’s & World Bank’s debt traps.

Why did the US reject Russia from joining NATO? Because the US doesn’t want Europe and Russia to develop closer ties, because it doesn’t want the “Eurasian landmass” to ever cohere, because then it would become too self-sufficient and powerful for the US to control. Zbigniew Brzezinski laid this theory out when the Soviet Union fell. That’s why the US tried to convince Europe not to build Nord Stream 2 and then later not to turn it on, why Biden said he would “bring an end to it” if Russia invaded, and why they ultimately did bring an end to it.

The US also very much wants regime change or balkanization in Russia so it can resume its neocolonial “shock therapy” plundering of it, which started under Yeltsin and ended under Putin. That’s why the US has a special hate-on for Putin.

Compare what Russia has done in the last 80 years to the US:

• List of Atrocities committed by US authorities
• A Detailed Chronological List of US Interventions, Invasions, Destabilzations, and Assistance to Oppressive Regimes (ending in 2002)
• The U.S. Did Not Defeat Fascism in WWII, It Discretely Internationalized It
• Shock therapy (economics)
• Are We The Baddies?
• The blueprint of regime change operations How regime change happens in the 21st century with your consent
• Infographic: US military presence around the world The US controls about 750 bases in at least 80 countries worldwide and spends more on its military than the next 10 countries combined.
• World Incarceration Rates If Every U.S. State Were A Country
• Michael Parenti: Africa is Rich

As for the US’ actions against post-Soviet Russia in particular:
The US has wanted to break up or otherwise weaken/isolate Russia ever since almost immediately after the break-up of the USSR. That’s why it’s been expanding NATO ever-closer to Russia despite originally having sworn up & down never to move one inch eastward. The US couldn’t allow a Ukrainian government to stand that was friendly with Russia. That’s why it couped Ukraine’s government in 2014.