Can I set up Wal Archiving on a three years old Postgres database

Yes, it will catch up, using streaming only, if (and only if), the number of WAL segments generated since the last update on the standby is less than the value of wal_keep_segments in postgresql.conf. This is covered in this section of the documentation: Replication

In order to restore a backup, you need to have the base archive of all the data files, plus a sequence of xlogs. An "incremental backup" can be made, of just some more xlogs in the sequence. Note that if you have any missing xlogs, then recovery will stop early.

So it's not clear here exactly what you've done, unless you changed the level of detail you're mentioning part way through your list. When you make a copy of more segments that have been put into the archive directory after adding more data, you need to ensure that all the data has been archived: using pg_start_backup and pg_stop_backup usually does this for you, but you don't mention it the second time. You need to at least do a pg_switch_xlog to have the current xlog segment immediately archived.

If you think that recovery is not consuming enough xlog segments, look at the recovery log to see if it tried to take them all. And have your recovery command make some sort of mark on which xlog files were taken.