I quote here what you have said:

"my understanding is that all stream ciphers leak some bits of the key with enough output"

This is a false statement.

Nowadays, the most frequently used encryption method is to use a stream cipher (more precisely a binary additive synchronous stream cipher), where the automaton that generates the encryption mask stream is implemented with a counter and with an output function that is an unpredictable pseudorandom function, usually either AES or ChaCha20.

With an appropriate pseudorandom function, such a stream cipher does not leak any bits of the key, even with enough output. There are methods of breaking a cipher with enough computation and memory, but those are specific to the pseudorandom function that happens to be used and they are not limited to stream ciphers, they are equally applicable to block ciphers that use the same pseudorandom function. Moreover, the known methods used to construct block ciphers are usually easier to break than good stream ciphers that use the same pseudorandom function.

It is obvious that when you have written "all stream ciphers", what you had in mind were not "all" such ciphers, but certain stream ciphers with very low implementation cost, but also with inadequate strength, which have been specified in many older communication protocols, e.g. in those for mobile phones (because the communication equipment manufacturers only cared about a low production cost, not about security, and the so-called encryption was only added to fool the customers, not to prevent professional attacks). Such cheap stream ciphers have been easily broken by anyone who has studied them seriously.

The ciphers that you have designated by "all stream ciphers" are an extremely small fraction of what is meant by "all stream ciphers" in the cryptographic literature, as I have explained in my previous post.

Your confusion might have been increased by hearing people referring to AES as being a "block cipher". That is very wrong, because AES is not a block cipher. It is not even a cipher. AES is an unpredictable pseudorandom function, which maps an encryption key and an input symbol to an output symbol. Alternatively, AES can be viewed as a parametrized family of bijective functions (a.k.a. invertible functions), where each of the bijective functions is identified by a parameter that is the encryption key.

With an unpredictable pseudorandom function, it is possible to build various kinds of ciphers, either block ciphers or stream ciphers. Nowadays it is much more frequent to use stream ciphers based on AES, than block ciphers based on AES.

General disadvantages of the block ciphers are that the message must be padded to a multiple of the block length and that both the direct and the inverse forms of the invertible pseudorandom function are required to be implemented.

In the past, because authenticated encryption was not used, block ciphers (e.g. the so-called CBC mode) were preferred, because they made difficult the tampering of the message by amateurs, even if, in the absence of authentication, they had little chances to stop a professional attacker.

After the use of authentication has become ubiquitous, the stream ciphers have become preferable for most applications, even if there remain a few niches where block ciphers have advantages.