I'd use Electrum or the iancoleman bip 39 tool to generate some seed words, verify that it works with Electrum, and use that for a paper wallet with the native segwit addresses. Offline of course, separate physical computer or laptop.
That's not what OP asked for.
The question was not how you would create a paper wallet, but how to transfer coins from an already existing one.
So if I understand correctly, my current paper wallet with old address format P2PKH and corresponding private key is save for spending in the far future (I'm planning to keep them many more years). The bitcoin protocol will always support this current paper wallet?
Yes, if nothing severe happens (i.e. some kind of vulnerability exploit), they will continue to be supported.
And if something like that would happen, this would lead to way bigger problems for the whole network and its users than just your paper wallet.
If you still want to transfer your funds, make sure to do so on a safe device.
The more coins you have on this paper wallet, the more paranoid you should be when typing that private key into a software/computer.