In a Serial 3-Server Replication Scenario (A>B>C), can server B preserve binlogs that server A has purged?
No problem.


A (as a Master) writes to its binlog
That stuff is copied (usually immediately) to the "relay-log" on B (as a Slave).
That stuff is written to B's binlog
That stuff is copied to the relay log on C.


If the network from A to B goes down for more than 10 days, you will lose data.  For B to C, you have 30 days to repair the network, even though some stuff may no longer be in A's set of binlogs.

Another thing to note:  I think the purging of old binlogs does not occur until the current binlog exceeds its limit.  If that limit is large (in comparison to the activity), it could take days to roll over.  You might want to shrink the limit to get better granularity.  (And it is OK if A and B have different limits.)