On the contrary, many of the thoughts that I have are ones that I do not endorse, although I am perfectly aware that they are my own.
The sentence you quoted lacks the main verb, which I added above.
"Ones" is correct there because it substitutes for "thoughts," with the zero article. You would need "the ones" if "the thoughts" were required:
- There are different types of ideas: ones (= ideas) that I endorse and others I don't (also: the ones I endorse and the ones I don't). The ones (= the ideas) I endorse are the ones in line with my line of thought.
"Those" is always interchangeable with "the ones." Notice that in my sentence above — though not in the sentence you quoted — "those" can be used because it is possible to classify your ideas between the ones (=those) you endorse and the ones (= those) you don't.
The difference between ones that I endorse and others I don't and the ones I endorse and the ones I don't is that the first classification is open (there may be a third group of ideas the person only endorses (or doesn't endorse) partially, while the second one is a closed classification.