US market I don't know about but for the ASX yes, you own the shares.
If the broker goes bust then you'll have some paperwork to sort out in selling them but ultimately they are your shares not the broker's.
I personally wouldn't use any brokerage arrangement that didn't give me ownership of the shares. That's an unnecessary risk in my view that there's no need to take given that there are plenty of brokers around who are indeed just middlemen, they facilitate the transaction but you own the shares.
The other thing I'll mention is price. What you want is first and foremost a broker that is reliable. It's no good having fancy features and so on if you can't actually buy and sell when you want to because their website can't cope on high volume days or whatever. Brokerage fees are important but it's more important to have reliability than a cheap price in my view.