The Ledger debacle that caused the leak of customer information (down to their home addresses) is making me hesitant in purchasing a hardware wallet.
There are ways to buy a hardware wallet without giving away your personal information. You can find an official reseller in your country and buy from a physical store using cash. You can have it posted to a PO Box or using general delivery, so you do not have to give away your address. You can also buy it using a pseudonym with delivery to a different address (if you have one you can use). And of course if buying online pay using well anonymized bitcoin and not a fiat method which is traceable to your real name.
Should I go for a paper wallet instead?
If you are planning to hold for a long period of time and transact very rarely, then a paper wallet can be a good choice. However, setting one up securely is significantly more difficult and has significantly more risks than using a hardware wallet, as does when you come to actually spend from the paper wallet. I would not recommend that your first attempt at making a paper wallet is then used to store the majority of your funds.
I would prefer to use an existing way[2] how to create paper wallets
The site linked to in the thread you have linked (bitcoinpaperwallet) is a known scam. Do not use it! Please also edit your post to remove the link to this scam.
Creating cold storage using Electrum is not secure as hardware wallet since you're still connecting to internet to download Electrum or you're connect it with your insecure device.
This is not correct. A properly set up Electrum cold storage wallet is just as safe, if not safer, than a hardware wallet. You download Electrum on a separate device, verify it, and then transfer it to your airgapped device using a USB drive. Your airgapped device remains permanently airgapped and the wallet on it never connects to anything, but simply signs transactions created elsewhere and imported using a QR code or USB drive.