>But then any sort of management information, eg "find out how many active customers we have" or "which customer has the most activity" becomes a query over all the DBs
Create a separate DWH storage where all necessary data from all databases is pulled and processed/aggregated in the background (via change data capture) and saved in a format that is more suitable for analytical queries. The downside is that it requires additional preparation and some upfront design.
>Similarly, admin tools for customer support etc easily become very hard to build.
Our CRM understands the concept of organizations/tenants, so when a customer files a ticket, we know their organization, and from there, for a tech support person, accessing "coolcompany.mysaas.com" is no different from accessing "mysaas.com". In my opinion, it makes tech support easier, because you have far less data/fewer logs to analyze/sift through (only a specific organization's DB) when investigating a problem.