I use Google Kubernetes Engine(GKE) to deploy my service. In the cluster, I enable Stackdriver Kubernetes Engine Monitoring instead of Legacy Stackdriver Logging and Legacy Stackdriver Monitoring. With the legacy monitor, I can find the metrics of the number of logs with the name log entries. What is the corresponding metrics name with Stackdriver Kubernetes Engine Monitoring?
If you go to Stackdriver monitoring > Resources > Metrics Explorer and select "Kubernetes cluster" as a resource type, you can find a metric called "log_entry_count" and select it. This metric is also mentioned here.
So - the metric you're asking about is still there - no matter if you create a cluster with Stackdriver Kubernetes Engine Monitoring enabled or no.
Furthermore - it will still collect data about number of logs ingested.
To be sure of the metric existence and if it actually does work I created a test cluster with some back-end service which generated some log entries and then tried "log entries" metric to count them - it worked as it should.