No. But Russia does believe they can outplay the US in regions where Russia has particularly strong interests. So far they've mostly managed to do that. The current Ukraine mess is of course a debacle, but until that Russia had made some serious gains on the international arena.
What exactly are you asking?
You know the opinions of Russian general staff, high command, and military scientists in the academy? Real ones, not the "Russian Stronk!!!" stuff for the media.
NATO does hold large scale exercises. Perhaps not as grandiose as the ones Russia has, but if you look closer at the details you will realize that while Russia has regained the ability to move and deploy masses of men and equipment, this ability is not nearly as solid or foolproof as you might think. And of course there are tons of lower-level problems from old equipment, to a badly trained NCO corps, to inadequate training methods and standards, etc.
For example Russian line infantry (Motor-Rifles) don't train for rapid-reload drills, don't know what a combat glide is, etc. Individual accuracy with personal weapons (AKs) is poor, and not just because the weapons lack optics or is inherently inaccurate (it's not), but because the standards for weapons training are low in general. The low-level officers and NCOs don't have the skills themselves, and thus can't pass it on to the soldiers. Contract soldiers, especially NCOs, are better, and quite valuable, but they're not nearly common enough (they're heavily concentrated in technical jobs that require more knowledge like SAM operators).
NATO is nowhere near falling behind Russia. If the Russian economy continues to develop successfully for another 10-15 years, and Russia continues to maintain a high level of military spending, with a competent and capable Ministry-level staff that understands how to get what it needs despite civilian pressures, then Russia may have a military comparable to that (in training, gear, etc.) of western Europe. But that's a big "if".
Also if you honestly think that the primary reason to avoid going to war with Russia, is because NATO is afraid they will lose, you've lost touch with reality. War is expensive, messy, and damaging. Europe has no desire for a major war. So the reason NATO is not going to war with Russia (the primary, if not only, reason) is that it's members don't want to. Even Britain has no desire for a large scale war with Russia.
