Verifying personal wallet keys - Hashing question

Hi all,

I have a question for the more mathematical folks.

I still like using paper wallets or simply having records of my private and public keys instead of hardware or software wallets, particularly as I see yet more reports of sophisticated people in the space having wallets with seed phrases compromised.

What does make me nervous about generating "paper" wallets is if you HODL a decent amount of funds there is always a moment where you hold your breathe and hope for the best when time comes to get funds off that wallet using the private key.

If I generate a new wallet, is there a way I can verify my addresses for confidence? i.e. can I hash the longer private key using a particular algorithm, whereby the result should be the shorter public key? Maybe it doesn't work like that and both the keys are derived from something else.

Can anyone guide me on this?

Thanks!