Answering my own question: My systems do zero-downtime deployment. Some of my services are managed using ECS and some using custom deployment scripts.

It's interesting that people mostly focus on the mechanics of launching the new code. To me, the interesting thing about zero-downtime deployment is what happens while the release is in progress, when there will be a mix of the old and new code versions accessing the same resources (databases, microservices, etc.) at the same time.

For example, you don't want to just drop a previously-mandatory column from a SQL database: even if your new release no longer references the column, the new code will break if you deploy code before updating the database, and the old code will break if you update the database before deploying code. Obviously there are ways to do this kind of thing (roll out the change in small backward-compatible steps) but they're extra work and can be easy to get wrong even if you're using ECS to launch the code. Whereas, if you're allowed to take downtime, you can do it all in one step without worrying about mixed-version environments.