You're right on all points, and "yes" to your question.
Automatics restrict how much freedom the driver has in choosing how much to load up the engine. (EG you can't short-shift an automatic.)
Also most automatics are ill-suited to being used for one of the most effective ecodriving techniques: engine-off coasting. (Most autos rely on an input shaft-driven pump for lubrication. With the engine is off, these can be damaged by excessive coasting.)
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