When you read English (or whatever your native language is, assuming it's alphabetic), do you read the letters individually and sound them out to form words? Of course not; long ago you achieved reading fluency.
Same with music - with enough practice you just look at the score and automatically play what's in front of you. It doesn't matter if one staff is in treble and the other in bass, or if both staffs are in bass, or if the top staff is in bass and the bottom in treble for some hand-crossing passage, or if the composer decided to write in 3 or 4 staffs to make the voicing more clear.
Just as there are debates (see phonics vs whole language) about the best way for children to learn reading English, there are debates about how people should learn to read music, but, just as for English, it probably doesn't matter that much in the long run.
If you start playing chamber music with cellists or violists, you learn to fluently read tenor and alto clef so you can follow what they're doing. Probably you can depend on your tenor clef experience for B-flat instruments (though the few measures Brahms wrote for clarinet in A-sharp are quite annoying).