I have tried to reproduce the problem with 3.3.8 and 3.3.10 but current module seems to work fine (at least it generates events). I am adding some tests to cover these versions.
I wonder if the problem can be specific of its usage with https or something related to etcd API version. In any case there should be errors in logs. Could you check if there is any error in metricbeat logs?
It'd be also useful to have the etcd parameters and the configuration you are using.
And if you have the chance, it'd be also great if you could try to reproduce the issue with etcd 3.2 and/or without SSL.