Addendum: I think "many times" as an adverbial in sentences like these implies countability. It doesn't simply mean "often" or "regularly." We can (in theory) count the yearning events of the past, and we can project that there will be a countable set of yearning events in the future. But in the present we simply take our yearnings as they come. All this became clear to me when I thought of one exceptional case where I think the simple present does work with "many times" in the example in question. The exceptional case is that of the habitual yearner. If one knew one's own psychology so well that one had identified predictable occasions on which one habitually yearned to be free, then one could felicitously say: I yearn to be free many times. / Many times do I yearn to be free. /Many are the times I yearn to be free. The sentence could be followed like this: I yearn to be free at Christmas. I yearn to be free on my birthday. I yearn to be free when I think of my loved ones. Etc.
