A hardware wallet is safer than a software connected wallet because it works in the same way as a cold wallet.
Not really. A cold wallet should be permanently airgapped and never connected to another device, network, or the internet. Most hardware wallets on the other hand are designed to be safe even when connected to an device with internet access.
That is to say, it internally signs transactions with its seed and only sends those signed transactions to the computer without exposing the seed.
Private keys sign transactions, not seed phrases.
If you want to safely store your funds in a mobile phone, it's better to use an old one as a cold wallet.
I don't think it is possible to turn a mobile phone in to a true cold wallet. Even on flight mode it still receives and transmits data via a number of channels.
You can also do the opposite : using an old laptop or tablet that you won't connect till their seed is in use
Cold storage should never connect to the internet, even for a moment. If it does, then it is no longer cold storage.