The US has to worry about letting advanced weapons systems fall into Russian hands, too, as they inevitably will under battlefield conditions.
Certainly they do worry about this, but whether they
have to is more questionable. The less advanced systems are technologies that Russia already has, and even those are proving difficult for Russia to manufacture and deploy in large enough quantities to make a difference.
More advanced technologies are simply beyond what Russia can manage to field economically - if they were to capture and reverse engineer them, that might well actually benefit Ukraine, because attempts to manufacture and deploy them would divert funding and effort from simpler technologies that would give a larger and more immediate benefit.
Hitler's V weapons looked spectacular, and were undoubtedly cutting edge technologies that outstripped anything their opponents could field. But the money, effort, and expertise wasted on those systems likely accelerated the German defeat - if they had spent the money on better winter uniforms, more fuel supplies, or better trucks to support the Eastern Front, rather than on whizz-bang secret weapons that were of zero tactical or strategic value, they might even have won the war.
Without the massive American technological and financial base that produces these advanced systems, they might well do more harm than good to anyone who tries to deploy them.
It's an integrated system - knowing how to make an Abrams tank with all its various secret features and capabilities is one thing, but having the resources and skills to build a fleet of them (including a supply chain producing high quality components on-time and on-specification), and the structures in place to test them, support them, train their crews, and effectively integrate them with other advanced weapons systems (including systems being operated by different armed service branches) on the battlefield, isn't as easy. Ukraine would presumably be able to lean on the US to help out with that stuff. Russia would need to do it all themselves (unless you think North Korea or Iran are advanced enough to assist).
This would be particularly difficult for a cobbled together conscript military whose doctrines and equipment are forty years out of date, backed by a kleptocratic industrial system that struggles to produce decent quality and quantity of even their relatively simple weapons systems and the fuel and ammunition they require.
The US military is a machine, and all its components are to some extent dependent upon all the others to make it an effective machine. You can put a supercharger from a Lamborghini on a Lada 2105, but it will still be a piece of shit.
Allowing Russia to obtain and examine such classified weapons systems is really not a big deal in the short to medium term. It could, perhaps, be problematic in the much longer term, if some future enemy (perhaps even a future, less comically incompetent, Russia) attains the ability to emulate the currently state of the art US systems; But by that time one might reasonably expect that those systems would be rendered obsolete by new systems anyway, at least for use against the US/NATO.