Alex, did you make up those examples?

Or were they maybe made up by AI?

No, I.took them from The Free Dictionary: https://idioms.thefreedictionary.com/at+(one%27s)+fingertips

They’re so bad 😅 we normally use this expression to talk about information that we can find very easily when we need it, not information that we’ve memorized or objects that we can reach. 

Ok. I'm going to revise my translation today as soon as I get time. How about this example from the same Free Dictionary:
"Because the Internet is so easy to access these days, we have every resource imaginable at our fingertips" ?
I don't know why, but I would replace 'because' with 'since'.
Ok?

I like BECAUSE here and it’s a perfect example 👍🏼🤩

By the way, Uly, a question about accessibility:
«You don’t have to say you love me, just be close at hand.» If you blindly translate the phrase word to word, «будь под рукой» would mean in Russian «be around, because I may need your assistance at any time.» We normally use «быть под рукой» to talk about objects in our reach, like books, tools, etc.
Is it normal to have somebody at hand meaning just a physical nearness?

No, I think they just used it in the song for rhyme, etc.

👍🙏🤓