You're right that the solution shouldn't be automatic US approval for any drug used in Russia. It's reasonable to have standards for drug testing and require that any drug used in the US is tested to those standards.
The problem isn't (only) that US drug standards are strict. The problem is that the financial burden of testing a new drug, which is significant, lies solely on the manufacturer, who can only recoup those costs by selling patented drugs at high prices. This means that the incentives don't line up for drugs not covered by US patents: because there's no profit in selling them, no manufacturer can justify funding the clinical trials to get them approved. This despite the fact that society would, as a whole, benefit from having those studies done.
One obvious solution would be a government program whose mission is to identify promising treatments for which private incentives don't line up, and shepherd them through the approval process by funding clinical trials and so on. You could also imagine a free-market variant, in which the government just auctions off US patent rights for foreign drugs; the auction winner gets exclusive rights to sell the drug in the US for some limited period of time in exchange for agreeing to fund FDA approval.