I've created a bech32 segwit paper wallet using this generator:
https://segwitaddress.org/ . Everything was done in a new computer, with a live Ubuntu OS installed on a USB stick, and no internet access of any kind. I got a private key instead of a seed.
I find it contradictory that you have taken precautions like using live OS without Internet access, but then use a Bitcoin software that wasn't verified neither by you or anyone else. I don't know if that site is legit, and you can never be sure that just because there are no scam accusations against it, it's 100% safe. We have open source wallets like Electrum or Bitcoin Core which are used by millions of people, why not use them?
Will my new cold storage wallet still be safe this way? Or should I create a new one which makes use of a seed? I'm used to handling private keys from many cryptocurrencies ever since I began my journey in 2015. I don't like seeds because they have a bunch of Bitcoin keypairs attached to them. A private key only works for a single address, allowing you to easily manage your funds.
That's the point of a seed, you have many addresses, so you have better privacy. It also removes other risks that are present with single-key paper wallets, like losing your change address or loss of the key due to partial damage. You can also write down your seed with just pen and paper, which means you don't have to introduce printers into the process of making a seed, and printers could be vulnerable, especially if you don't own one.