TL;DR version:

Other answers have commented on workarounds and use of the more recent Traits options. However, none quite tell you how to specifically exclude tests for a trait. To do so, simply use a - (minus) to negate a filter in the search box, e.g.:
-Trait:"DatabaseIntegration"
This will exclude all tests with that trait. The MSDN documentation on these features has the following explanation:
To exclude a subset of the results of a filter, use the following syntax:
FilterName:"Criteria" -FilterName:"SubsetCriteria"For example,
FullName:"MyClass" - FullName:"PerfTest"returns all tests that include “MyClass” in their name except those tests that also include “PerfTest” in their name.