Seeking key stream as way to reuse a key
To address your bullet point directly, yes this will work for all generic key streams that only produce a sequence dependent solely on a key. 
 However these are some real world problems.


Your scheme requires perfect synchronicity between a single pair of sender and receiver.  Otherwise the stream position will get lost as that is not part of the transmission.
You've specifically mentioned attackers.  Receipt of an attackers message will throw out the stream position and you'll never recover it without careful tracking of the number of characters received. Perhaps not even then. 
Similarly, anything other that a single pair of sender /receiver will make stream position recovery extremely difficult. In the long term, impossible /impractical.
Salsa20 /ChaCha20 overcome this problem by use of a nonce that resets the key stream to an identifiable position.
If you don't exchange keys, how can new members join the party?


Key exchange is a part of cryptography and is well addressed.  Key stream positioning is the role of a nonce, and well discussed.