Thus it is possible to transmit arbitrary (classical) data with exponentially decreasing probability that we do not detect an eavesdropper. Alice and Bob can communicate with each other knowing that there is provably pretty much no chance that anyone else knows what they said to each other.

As stated, this isn't possible with encryption enabled by quantum key exchange, either, nor is it an intended goal of the procedure. Quantum key exchange empowers integrity in the cryptographic key exchange process, but it doesn't intrinsically do anything to empower confidentiality (encryption) or authentication. If you successfully share a key with another party using a channel established via quantum entanglement, you will be (theoretically) capable of discerning with certainty whether or not there has been an attempt to record the key "outside" the channel.

The quantum innovation ceases once the key has been successfully shared. The ciphertexts encrypted with a key shared via quantum key exchange are just like ciphertexts encrypted with a key shared via traditional key exchange. They will not "break" if observed by an unauthorized party; in fact, there is no way to endow a ciphertext with that property. Either a party has the correct key or the don't, but the ciphertext will not "self-destruct" or cease to become usable if the incorrect key is used.

I feel this is an important nit to pick because even if you understand this, others reading your comment might not. Conceptually speaking, assurance of secure channel integrity is very different from assurance of confidentiality. It would be more accurate to say that in a quantum channel, you could exchange information in such a way that it cannot feasibly be read or tampered with; however, if the ciphertext were extracted from that channel, it would not be meaningfully different from a ciphertext extracted from TLS.

Now with how complicated and costly creation of the entangled particles will be, I'm assuming that usage like this will be very very rare, but it's still possible.

Sure, I'm not saying you can't achieve information theoretic security. My point here is that discussion of quantum key exchange should use precision in terminology - integrity and confidentiality are meaningfully different. If the ciphertext from a quantum channel is extracted (in whatever way), it is not more secure than a ciphertext in which the keys were shared in person and promptly destroyed via Cold War-era means. You won't be aware of an adversary trying to break the encryption once they have it in their possession, and the ciphertext won't "break" if plied with an incorrect key (though it would be secure for other reasons). Rather, you'd know if the key exchange process is being broken, or if information was not transmitted correctly.

Edit: This kind of observation would already require physical access to the communications medium (e.g. a fiber optic cable). So if you could DoS a quantum encrypted channel by measuring it, you could equally well DoS a classical channel by just cutting the fiber.

The comment just above mine, which does a better job than me of explaining the degree of access required.

Besides which, it’s not as though you can only use this. You try your quantum channel first, and if it’s down you know you’re under attack, and act accordingly with your Classical backups.

Replacing that with a system where your adversary can't see your message, and neither can your correspondent, is a downgrade, not an upgrade.