Distributing the database to customers is still workable despite this drawback - antivirus systems are a common example.
I don't agree with that. Having the database only allows bad actors to see if they've been added to the database - not the business logic of how they ended up in that database.
You could argue that knowing you've been added to the database is still an advantage, which is true. But those same bad actors could still test the system with SaaS solutions too. eg when one acquires stolen bank cards it is common to use those details to make small payments for things like food delivery to test if those card have been blocked or not.
I wouldn't say it's preferable for the customer. It's just one of many options.
> but it's obvious why a vendor wouldn't be keen
Yes, SaaS obviously has benefits for vendors too but what's most preferable to them is making money. If every customer refused to do business with SaaS then vendors wouldn't prefer providing SaaS solutions (yes, I know I said "solutions solutions" :P). The reason SaaS is so prolific is because it offers advantages to customers as well.
When has SaaS stood for anything besides Software as a Service? Because software as a solution could describe almost any business model?
The obvious downside to that is you don't have real time updates (as well as the usual other benefits that SaaS brings), which is why SaaS suites this kind of business well. But it can be done and was sometimes done this way "back in the old days".