Is a CNA course enough experience?

It's impossible to generalize. It is more difficult to get a hospital job than LTC and of course, there are many times more LTC jobs in the first place.

The hospital I worked at did take new CNA's and put them through a training program. But, they had a fairly rigorous interviewing and testing process. They were more interested in literacy, honesty, background in school and grades, and self-presentation (meaning how well you did interviewing) because they were extremely customer oriented.

I have posted a number of times about interviewing as I have vast experience doing this (in multiple careers). I won't post anything here other than to say that you need to be sure you present yourself well, write a good resume, fill out job applications legibly and show you know spelling and grammar. Go for it!

As far as nursing schools, one of the largest reasons there is a shortage of nurses is not enough schools and people who want to teach. Waiting lists are all long for all schools. Again, as a generalization, you need to have top grades in both high school and as a CNA in order to compete. That's what will get you hospital experience too. And again, go for it. You have nothing to lose by filling out the applications.

You can also spend a year at LTC getting solid experience and try again for hospital work, if that is what you want.