Ironically enough, your post is exactly the first lesson Luke was trying to teach Rey. The Force doesn't belong to the Jedi. Having a connection to the Force isn't the sole purview of the Jedi.
Even Yoda says this in Empire, and Obi-Wan in A New Hope. The Force resides in all matter, and in between all matter. And some people have an affinity for accessing it compared to others. The Jedi were specifically looking for people that showed a high affinity for connecting with the Force.
Han doesn't connect with the Force not because he can't, but because he has no interest or desire to try. Luke fails to lift his X-Wing out of the swamp, not because he can't, but because he doesn't think he can do it. So no, Han couldn't have become a Jedi specifically, but he's still a part of the Force, and possibly had Force potential if he was open enough to tap into it and let it flow through him.
Being a Jedi or Sith isn't the sole purpose of the Force. What Luke was teaching Rey (and the audience, who also share her misunderstanding that only Jedi and Sith can wield the Force), is the Force is more than just the Jedi, Sith, and lifting rocks. Though Rian humorously has her lifting a shit ton of rocks at the end of the film, which I loved.
Luke wanted to end the Jedi because people's insistence that the Jedi way was the only way to resolve conflict or to engage with the Force is flawed. In a way, it parallels modern religious conflict. I was raised Christian, but it's foolish to think that it is the only path in life, or the only ethic standard by which people should live their lives and find meaning and purpose, and that those that either follow a different faith, or don't believe in any faith at all are wrong or bad people.
Luke did come to realize that the people's adulation of the Jedi and the idea of the Jedi could prove useful in galvanizing them to reach into themselves and find the strength to fight against the totalitarianism of the First Order in their own way. Much like Poe coming to realize that hopping in a ship and blowing things up isn't the only way to be a "hero," or to lead and inspire people.
You don't have to be a Jedi to access the Force. The Force is in everything. It belongs to everyone. Much like Star Wars.