Yes, my confusion was that I thought those wallets could be used only as hot wallets. I knew they are software wallets but it never occurred to me that I could use them "offline"
That's OK. When Electrum is used in an offline/airgapped environment, you don't have to fear that your private keys will leak. Assuming, of course, the computer you use is clean and malware free. The device should also be formatted and not be connected to the internet after a clean install of the OS ever. Most people recommend using Linux, but this is your choice. It's also recommended to remove ethernet and wireless cards from the motherboard so no one can try any tricks. Full drive encryption is another noteworthy step.
After that is done, you use the offline machine only for constructing and signing your transactions. The broadcasting takes place on a different online computer. The signed and unbroadcasted transaction can be transferred for signing using a USB drive or via QR codes.