Colin the Bear couldn't have said it better.
I used to (and still do have) some troubles with my sight reading. I learned to play by ear and so memorised a good number of tunes.
The way I think about it is like telling stories. Some people are great at reeling off stories from memory. Other people can read really well. Then you've got some people who can read off the page and still make the story jump out and come to life. I think without being able to read music, it's like loving great stories without being able to read books. They both have their place and use, but one definitely complements the other.
Having a good ear for a tune or the ability to play songs from memory helps when reading new tunes. As I read the tune, my mind and ear are searching for the tune or shape of the melody, which helps the piece flow better in my head.
Although you didn't mention it in your original question, others in this thread have mentioned knowing the chords of a song. I didn't think was so important when I started playing sax, but onceI wanted to learn how to improvise properly, it gave me a framework and structure. A great teacher I had once made me play all the arpeggios of a tune I thought I already knew. Suddenly I felt the shape of the song as well and it was a great help in being able to make more informed choices when improvising.
Going back to your original question - whether learning tunes should be a goal in itself or will just come as a byproduct of loads of practice, I would say both. If you play a song long enough, you will remember it. However learning tunes exercises different musical muscles.
If you can hum it, you can progress to working it out using just your ears. In jam sessions or where you don't know the tune, being able to pick up the key tones/chords quickly is just an extension of trying to work out a tune that you hum/hear.
Good luck and I hope that you feel you're question has been answered by all the other quality answers above!