Removing the "LIMIT 10000" would cause the service's performance to degrade horribly for everyone, and most likely none of the searches for very large result sets would actually complete before being killed off by the database replication.
The identity searches are powered by PostgreSQL's Full Text Search. I've not yet(*) been able to find a way to use Full Text Search as part of a composite index (on e.g., issuance date), so adding date filters isn't as simple as you might think.
(*) This is just a half-baked idea in my head so far (unfortunately $DAYJOB hasn't afforded me much time to spend on any crt.sh projects recently): The "certificate" table currently has a partition for each calendar year (and each certificate is stored in the partition that corresponds to the certificate's year of expiry). I've read that newer versions of PostgreSQL can handle huge numbers of partitions, so I'm wondering about changing the "certificate" table to have a separate partition for each day. Since each partition has its own Full Text Search index, my thinking is that this approach might help to provide a fairly crude but hopefully effective date filter to accommodate large result sets.