Many versions of the schema: indeed, that's one of the main tradeoffs. I consider this to be a feature, not a bug, in an enterprise application context. The correct version of the schema that corresponds to a given version of the code is unique, which dramatically simplifies debugging and migration.
Separate backup scripts: only if you want, you can always backup the whole database server/instance at once (i.e. an RDS snapshot). On the other hand, you can do more fine grained backup controls if you want to (and with enterprise customers, you likely will). Need short data retention for an EU based healthcare provider? Good luck purging that data out of your consolidated cluster-wide backups.
Busting query cache: I don't think this setup is detrimental to query cache functionality, unless you have many queries shared between tenants, which you generally don't in this type of setup (i.e. frequent queries like "SELECT * FROM post_categories WHERE tenant_id = XX" is not a cache entry that can be shared between tenants anyway).
Busting connection pooling: not necessarily, but if you want separate DB credentials for each tenant, yes. If that is not an acceptable tradeoff, you can reuse connections, at the expense of lessened data isolation between tenants.
Connection to the proper database: I would argue there's nothing uncommon about connecting to a specific database, but I'm not sure I see your point. You just issue a "use XX" statement before you start querying, or connect with different credentials if not reusing connections.
Collection of data across tenants: yes, cross tenant analytics gain overhead and complexity, but those are typically not part of the application code anyway because most of the time you need to manage separate clusters anyway (region specific, fault tolerance, dedicated, etc).
Anomaly detection at the database level becomes more complex, but (a) in my opinion, it's the kind of custom problem that you will custom solve anyway (so no huge cost savings) and (b) a lot of it happens at higher layers (New Relic, etc) which are not impacted by this architecture choice.