pg_upgrade would have worked fine given your requirements. Just using normal streaming replication to move database over to new systems and then performing an in place pg_upgrade there would be doable with most likely a couple of minutes of downtime and a much quicker and more robust process.
How would that have worked with multiple replicas cascading from the new primary? Streaming replication doesn't work across versions, so would we have had to build out a tree of new instances, then pg_upgrade them all at the same time?
My experience was that it needs to rewrite all the data for some tables/indexes under some circumstances, and the db won't be available while that happens. So, unless your db can be down for the time it takes to rewrite all that data it isn't really an option. After having done the upgrade through streaming logical replication I'm not sure I would try pg_upgrade again.
I did the pg_upgrade style update a long time ago, so most details are fuzzy, but I remember setting up a string of replicas something like:
primary -> [read_replica, backups_replica]
read_replica -> [upgrade_replica]
upgrade_replica -> [read_replica_upgraded, backups_replica_upgraded]
This allowed us to do multiple practice rounds without putting any unnecessary load on the primary. I think we needed to re-initiate replication off 'upgrade_replica' after the upgrade, but we did the live update during low-load so the extra read load wasn't an issue.