using a Ledger with a 24 word seed phrase just seems antiquated, like I said before. Is this really state-of-the-art?
It might not be state-of-the-art, but it is tried and tested, and known to be safe and secure.
The 24 word seed phrase is only used as a back up. Provided your Ledger device is never lost/damaged/stolen/etc., then you never actually need to use your 24 word phrase. You just need to know you have it in case of disaster. And it is much safer for your back up to have never touched an electronic device and be stored solely on paper.
The latest Ledger device works on computers and on mobiles via cable or Bluetooth. It supports hundreds of coins and multiple wallets via passphrases, and most importantly, it wipes itself if you enter the PIN wrong 3 times (which is when your 24 word phrase is needed to restore). I assume the facial recognition wallet does not have such a feature, and would let an attacker have endless attempt to bypass its security. The Ledger does everything that is needed to fully control your coins whilst also keeping them safe at the same time. It might not be as "fancy" facial recognition, but honestly, that's a good thing.
As Pmalek says, you should absolutely not be leaving your coins on any exchange. If your coins are on an exchange, then you don't own them - the exchange does, and there are hundreds of stories of exchanges disappearing, locking accounts for arbitrary reasons, or being hacked.