No, not like Mt. Gox.

The "second layer system" that people are actually talking about is Lightning Network, the basic idea of which has been around for a few years now. Basically, you open a "payment channel" of X bitcoins with someone else in the network, and you can trade up to X bitcoins with them almost-trustlessly - cheating is easily detectable within the protocol, and results in the party detecting the cheating settling the payment channel on the blockchain. Payment channels can be linked, so you can send money from A to B through payment channels A-X, X-Y, Y-B, without actually trusting any of those entities.

So blockchain transactions only ever happen when someone wants to open a new payment channel (which is only relatively rarely), when someone they directly have a payment channel open with attempts to cheat, and in the rare occasions that someone wants to hold "actual" bitcoins rather than a cryptographic promise of bitcoins.

There's no entity within this system which has central control over currency that goes through this system, and all cheating is detectable with rollbacks in place when it happens, so it's not like a bank, exchange, or even a traditional payment processing network at all.

Note: I have my own qualms with cryptocurrencies and the human layer, but you've got to admit the technological backing is sound within reasonable parameters.

Mt. Gox was a centralised custodian. Layer 2 systems like Lightning Network are not, because they can't run away with your money. If they try to cheat, you can always withdraw your money.