Is there a way to speed up the replication or should I rather stick to streaming replication? As I have only 1 database on the server, it would not be a show-stopper.
You have a method that works, and a client that is already twitchy about downtime and only upgrades their database once every 5 years.
I would not even consider the idea of combining a major-version upgrade with a complete change-over in replication technology in a single step in this situation.
If you will get some kind of benefit from switching to logical replication, you could first upgrade production and get a new physical replica going, then once that is returned to production you can create a new logical replica and get it all synced over at your leisure, then get it all tested and then cut the clients over from the physical replica to the logical replica.