I think it's likely that a fair number of native speakers share your belief that "bachelor's program" improperly suggests that a bachelor "owns" the program. That's probably why "bachelor program" is sometimes used, even though "bachelor's program" is more common.
My own take on this is different. When I use the phrase "bachelor's program," I am not suggesting "ownership" of the program at all. (If I were, I'd use the plural possessive and write "bachelors' program.") To see what's really going on, it might be helpful to start with the following two sentences:
He got his bachelor's degree from Harvard, but his doctorate from Yale.
He got his bachelor's from Harvard, but his doctorate from Yale.
These two sentences are equally grammatical and have the same meaning, though the first is slightly more formal than the second. The only difference is that in the second sentence, the word "bachelor's" is being used as shorthand for "bachelor's degree." Note that "bachelor's" retains its possessive form, even though it is now serving as the object of the verb "got."
Now consider this sentence from an article on a website called Coursera
How important [face-to-face interaction] is might depend on certain factors like your major and what sort of experience you want out of a bachelor's degree program.
Here, "bachelor's degree" is an attributive noun phrase modifying "program." That's perfectly grammatical, and the phrase "bachelor's degree program" is quite common. (In fact, Google's Ngram Viewer* suggests that it is more common than either "bachelor's program" or "bachelor program," at least in print.)
However, "bachelor's degree program" is also somewhat unwieldy, and it becomes increasingly awkward with repetition. Probably for that reason, the article's author occasionally shortens the phrase by omitting the word "degree," as in
The average public online bachelor's program tuition costs $38,496 for in-state students and $60,593 at private ones, according to US News—that’s total, not per year
Students typically need 120 credits to graduate from a bachelor's program in the US (roughly 180 credits at a school under a quarter system).
Here, since "bachelor's" is directly followed by "program," it may appear that "bachelor's program" refers to a program that is "owned" by a bachelor. However, I would argue that in fact, what is really happening is analogous to what we saw in our earlier example about the fellow with the degrees from Harvard and Yale. That is, the "bachelor's" in "bachelor's program" is shorthand for "bachelor's degree." As such, even though it retains its possessive form, it is really functioning not as a possessive but as an attributive noun modifying "program."
At least, that's what's going on when I use the phrase "bachelor's program" myself, and also how I understand it when I see and hear it used by others.
As to the question of which form "should" be used, it's really your choice. Both "bachelor's program" and "bachelor program" are fairly common, both can be justified grammatically, and both will be understood. "Bachelor's program" is more common, however, and might be the safer choice for that reason.
*If the graph does not display when you click the Ngram Viewer link, try hitting the return key on your keyboard.