The CDC for delta live works fine for delta tables, as you have noticed.  However it is not a full blown CDC implementation/software.

If you want to capture changes in Snowflake, you will have to implement some CDC method on Snowflake itself, and read those changes into Databricks.

There are several approaches to this, like using Snowflake Streams
or a commercial CDC software.

Depending on your scenario, you will also have to put some event queue between Databricks and Snowflake (like Kafka or Pulsar or ...).