MySQL doesn't store metadata about binary logs (well, it does, but only to control current and/or executed commands, not about the binlog themselves).
You can use
SHOW BINLOG EVENTS to read the actual binlogs, but that most likely will not be helpful, and will be very slow (it just read the logs). Yes, the only way I can think of is to read the actual logs with
mysqlbinlog and check the second line after the log position of the first and last events.
# at 256
#181130 9:28:36 server id 123 end_log_pos 545
If you don't need exact dates, there may be a shortcut for avoid reading the whole log- you can use the mtime (last modified time) of the filesystem metadata to get an aproximation of when it was last written (but not that will not be 100% acurate- flushing may be delayed by the filesystem, and the time the log was written may be different than that of when one the transaction was executed. But it may be a shortcut if you just want to find a particular event before reading just the one you are intereted in.