In Kibana > Stack Management > Index Management, with just a Fleet Server agent enrolled and running, only the metrics-elastic_agent.elastic_agent-default has a specific index template attached to it:

The other ones just have the generic logs or metrics index template:

This is because the Elastic Agent package only ships elastic_agent data stream:
https://github.com/elastic/integrations/tree/d7e83a6aa3b94cb9a4a94c205c0c580ace6a7cee/packages/elastic_agent/data_stream

(also, the elasticsearch.index_template setting should be removed as it is invalid)

title: Elastic Agent
dataset: elastic_agent.elastic_agent
type: metrics
-elasticsearch:
-  index_template:
-    mappings:
-      dynamic: false

It should ship all the other data streams too so that Fleet can install index templates properly for all of them.