How about checking if the number is computable?

It doesn't seem to me that the question is well posed, because you haven't given a precise notion of "test" and how a real number is given to you. My belief is that most explicit reals appearing in number theory are computable and thus inside a set of measure zero. See this previous answer by Joel David Hamkins for some of the hierarchies one can consider this way.

Notice the similarity that computability is also a property defined in terms of an approximation-by-rationals property, but even though it gives a way to recognize way more transcendental numbers, it is unlikely to have direct applications in number theory for the reason mentioned above. The results from Diophantine approximations you mention are, on the other hand, more valuable since they have applications ranging from solutions to Diophantine equations to transcendence of constants arising in various places (still within the computable world). See also this answer by Greg Kuperberg to a previous question, which is close to this point of view.

I'm sorry if this long comment isn't very enlightening, but maybe you can edit your question a bit to express what you would look for in a helpful answer.

How about checking if the number is computable?

It doesn't seem to me that the question is well posed, because you haven't given a precise notion of "test" and how a real number is given to you. My belief is that most explicit reals appearing in number theory are computable and thus inside a set of measure zero. See this previous answer by Joel David Hamkins for some of the hierarchies one can consider this way.

Notice the similarity that computability is also a property defined in terms of an approximation-by-rationals property, but even though it gives a way to recognize way more transcendental numbers, it is unlikely to have direct applications in number theory for the reason mentioned above. The results from Diophantine approximations you mention are, on the other hand, more valuable since they have applications ranging from solutions to Diophantine equations to transcendence of constants arising in various places (still within the computable world). See also this answer by Greg Kuperberg to a previous question, which is close to this point of view.

I'm sorry if this long comment isn't very enlightening, but maybe you can edit your question a bit to express what you would look for in a helpful answer.