Validate datasets are found when checking bigquery (#22144)

* Validate datasets are found when checking bigquery

Fixes #19709

* Address PR feedback

Made a general mechanism to pass expected messages to users in
api/database via ex-info. This allows us to suppress logging for
"unexceptional" exceptions that one can expect to hit while setting up
drivers.

* Only validate when filters are set

Also removed the dataset list from the exception as it's not surfaced to
users.