I don't think there have been any new designs for a good while that didn't. RC4 is, I think, the last actual stream cipher that saw widespread use (apart from designed-as-broken telco crypto). The same observation can be made for classified crypto, where they used to use bit-stream ciphers like Walburn or Saville (efficient, relatively low gate count hardware implementation) but pivoted to block ciphers since the ~80s.
More generally, I think the distinction between a "block-based stream cipher" and a block cipher (or permutation) mode is the degree to which you abstract away the block cipher. Personally I wouldn't put them together. Counter mode completely abstracts away the cipher, assuming it is an idealized pseudorandom function; sponge-based modes also abstract away the permutation as ideal in most cases; stream ciphers usually do not do this, and use weaker (but still sometimes block-based) primitives to generate output. There are, for example, sponge modes that use very few rounds past the initialization, which makes them more streamy than spongy. So their cryptanalysis is often different.
Just to make sure I understand you correctly, for example:
- CTR: definitely a stream cipher, and a synchronous one at that
- CBC: maybe a little like a stream cipher (you're e.g. exposed to the block width, and also less like most stream ciphers because asynchronous)
- AEZ: extremely not (because you're exposed to round counts)
That makes perfect sense to me, since that's where I felt the analogy started getting tenuous :)
Every entry in CAESAR that I've studied is either based on a block cipher or a hash function. A lot of them use AES internally, but others (MORUS, NORX) used their own designs.
There are no RC4-esque stream ciphers being taken seriously these days, for good reason.