Who are the most beloved pro athletes who weren't particularly great at their sports?

Vince Papale, Philadelphia Eagles.

Here's the plot of the movie Invincible from Wiki (although there are key differences between the plot of the movie and reality, this fairly sums up his story)

To spice things up for his organization, newly-hired Eagles head coach Dick Vermeil decides to hold open tryouts in 1976 for a spot on the pro football team's roster. Vince Papale is a 30-year-old bartender who has just lost his part-time teaching job. He had gone to Saint Joseph's University (then known as St. Joseph's College) on a track scholarship. Disgusted with his professional failure, his wife, Mellie Faircwib, abruptly leaves him, taking all of their furniture with her and leaving a note saying that he never will be anything in the world.

With the support of all his friends from the neighborhood, Papale goes out for an open tryout practice; catching the attention of coach Vermeil, he is invited to come along to training camp. Papale eventually makes the team, and after a dismal display in the opening game of the season, he recovers a fumble and returns it for a touchdown in the second regular season game versus the New York Giants, which leads to Vermeil's first victory with the Eagles.

Papale wasn't that great but, like the main character in many sports stories, that's what makes it interesting. Common guy, comes in and gets a shot. Plays well enough to do OK, but is never going to make a Pro Bowl or anything. That's not the point. He's a human who gets to play amongst the Gods. An inspiration for us all, etc.
Jonathan Brill
Written Dec 7, 2011