NASA could have sent a probe that sent a signal...

On this I actually agree. A disposable probe with only flyby capabilities and no need for reentry navigation - well within the technology of the time. It flies off, plays its tune, burns up in atmo on the way back, and the ships "find" the splashed-down crew module at the designated location.

That said, they'd still need pinpoint script timing to pull off the impression of real-time conversation between the pre-recorded messages from the probe, and the responses of mission control. And, of course, rely on absolutely nothing happening during the inevitable signal loss when the probe is behind the Moon. The fact the Soviets sent a robotic lander the very next year, speaks highly of the remote control technology of the time, at least for tasks as simple as driving a glorified RC buggy on flat rock.

But that's the thing - with automated technology being that advanced, it would cost nothing to strap on a couple of human passengers anyway, at least for bragging rights. Life support issues are the main concern, of course, but most of the conditions would be the same as in orbit - and those were already covered ground by the time.

And of course, this still leaves the smoking gun - terrestrial telescope images of the landing sites. Either put by hand or by machine, the one thing certain is that something manmade was placed there within the past fifty years or so.

As for the Ukraine conflict, Russia has already announced it will suspend joint operations in the ISS until the sanctions are lifted... however that is supposed to work, since the hub and life support modules of the thing are Russian. Either it's a token announcement with no practical substance, or, cynical as it may be, a testament that matters of space always have and will continue to be guided by terrestrial concerns, at least for the foreseeable future. So there's that.