Can't edit a feature class when there are no templates

Overview:

For editing purposes, feature templates appear in two places: the MXD and the Service's REST endpoint. The message "No templates available" in the viewer means they do not appear in the service's REST endpoint.

Solution:

To confirm this, visit your layer's REST endpoint in the browser. The URL will be something like:

http://sampleserver6.arcgisonline.com/arcgis/rest/services/Recreation/FeatureServer/0

From there, you should see templates.

For the Recreation layer, the templates are contained with in the description for the Types. If you don't have any Types, then the templates appear below the Fields in the interface, like this:
Templates: Name: Spills (Polygon)
Description: Prototype:
FIELD_DESC: null
STORY: null
Drawing Tool: esriFeatureEditToolPolygon

If you don't see any templates anywhere in the REST endpoint, open your MXD document in ArcMap and make sure that the templates you created are still associated with your layer. You can check this from the layer's context menu, under Edit Features -> Organize Feature Templates...