Warning - incoming long rant.
I agree with @turf3. The title of this thread is a false dichotomy. It has arisen in our culture for two reasons. One, the rise of rock ‘n roll, which encouraged the attitude that you had to hear it to play it (not a bad idea), but also that learning theory and notation and all the traditional skills was antithetical to the rock ethos. Two, the slow but inexorable removal of musical training from public schools.
Today, young people who want to play music either have to live in an area where there is good musical training readily available, or join a band and learn by ear. (Or get a DJ rig and push buttons…) Thus, there are those lucky few who learn at an early age to read music, Or those who have to figure it out on their own.
This is the source of this dichotomy - a lack of a cultural stress on schooling in the arts.
@Wade Cornell, no disrespect to you, you are a fine musician, but your rather incessant emphasis on the superiority of learning by ear vs learning by reading music is nonsense.
When we were both kids, back in the ‘50s and ‘60s, every child in the US was presented with the opportunity to study music in school. This was also true in some other countries (still is in some cases). We were taught the traditional way, learning to read music while learning to play an instrument or sing. The two went hand in hand.
I do not believe that this had any ill effects on those folks who ended up wanting to play music which did not depend entirely on notation. For example, jazz and its derivatives, such as blues, rock and pop. ( Well maybe the blues is a cousin rather than a derivative.) Skilled musicians, both professional and amateur, could read music AND improvise / play by ear. Almost every working professional in the jazz age was able to read music.
I do know a few self-taught musicians who cannot read, some of them quite successful. All of them consider it a lack in their training. Some end up learning to read to improve their employability; a notable example is Glenn Campbell, who came up as a back woods country guitarist, learned to read so he could get studio work, and ended up as a big star because producers and recording industry bigwigs heard him singing while he was just a grunt studio guy. (Not really just a grunt of course, he was a member of the famous Wrecking Crew.)
Reading music notation is just part of being a musician. For some styles of music, and particularly for us saxophonists, being able to improvise and learn tunes and lines by ear is just part of the deal. It is a bad idea to rely on just one of these approaches to playing music. Doing so places limits on your ability to fully experience the joys of making music.
Bottom line - learning music by learning to read is not an impediment to becoming a good “ear player”. Or vice versa. Like any skill, both require aptitude and hard work. Yes, there are great musicians who can’t read. Yes, there are great musicians who cannot play by ear. But there are way more that are skilled in both categories. Learning both approaches only widens one’s horizons and opportunities.