Thank you very much for the hint. I have looked into Zkool and installed it. I have an old Nano S and a new Nano X. Unfortunately, Zcash Shielded is not available for Nano X, I can’t install it.
It’s available for Nano S though, so I tried using it on my Nano S and installed Zcash Shielded there. I found a way to create a new Ledger-based account in Zkool and tried to create one for my Nano S. The first 2 times I got some errors in Zkool, but then it suddently said “please wait while we create the account, confirm on your Ledger device”. The device seemed stuck, but a few minutes later it asked me to confirm sharing the public key or something. I approved it, then approved another such sharing. So it seems to work on the device side, although extremely slowly. Then Zkool itself seemed stuck adding account, but after I clicked somewhere (or restarted the app, don’t remember exactly) I noticed that my account was added and one of my UTXOs was discovered. Then I tried sending it, and surprisingly it worked with Nano S. My second UTXO wasn’t discovered, so I re-added the account and this time it was discovered as well. In the end I managed to make another transaction sending the rest of my ZEC. So I was able to recover all of my ZEC without entering my seed phrase in any hot wallet. Thank you very much again. It’s a pity that Ledger website only points to an old tutorial that doesn’t seem to work, while Zkool does what people really need.
Ironically, I bought a Nano X to be able to transfer my ZEC, but ended up using a way that DOES NOT work on Nano X and DOES work on Nano S.