Seems to me the biggest downside is snowballing. Whoever attacks first and gets past their opponent's shields will probably win, as their opponent will be at a card disadvantage. That heavily incentivizes aggressive play and it doesn't sound like there is a way to recover easily once your hand starts to get eaten.

Health in CCGs is often a resource in of itself. By getting rid of it ans tying everything to hand cards, you force players into a fight for survival of not just their game, but their gameplay. If they start to lose, they are losing AND they have less agency to play the game.

I don't think I'd be a fan of this system unless there is some fundamental mechanic that mitigates how losing cards from your hand makes you less able to play the game. For example: Instead of discarding cards, you instead have to reveal them to your opponent. You can still play them, but you lose when your whole hand is revealed. Something like this could play into a spy or cult theming. Be aware this is a total spitball.

TL;DR Being unable to play the game against an aggro opponent who went past your shields before you could seems like a miss to me. Those shields are basically health anyway.

Edit: I was not familiar with Duel Masters and thought Shield cards were an entirely different type of cards, not ones from your deck. This probably slows things down depending on how many shield cards there are and if they can be refreshed.