Part of the reason I would like to re-apply is that I do not want to feel like a failure down the road, which is admittedly not the greatest reason.
"Feeling like a failure" when you got admitted into a good PhD program is absurd. Look at it from a broad perspective: only around 2% of the population in the US have a PhD, it's clearly not rational to consider yourself a failure when you're in the top 2%. This feeling indicates that you are a perfectionist person, which is common and a great quality in academia... unless it's excessive and leads you to being constantly unsatisfied with yourself, undermining your motivation and eventually the quality of your work. It's good that you're aware that this feeling is not a good reason to re-apply, before any decision make sure that you completely get rid of this feeling: it's irrational and has nothing to do with where you do your PhD, because what matters in a PhD is the quality of your research. Don't let this kind of feeling grow, because a PhD is already challenging enough for one's sanity.
However, I think that it does not have the same name recognition of US universities or professors that are exactly specialized in my area of interest.
There are lots of unknowns when one starts a PhD, and it's human to try to find certainty in objective facts such as the recognition of the institution. Unfortunately it doesn't work this way: the recognition you will get after the PhD has very little to do with your institution, it will depend almost only on the quality of your own work. A major factor for this is the compatibility between you, your topic and your supervisor. It's true that top institutions tend to have good academics, but this compatibility is not automatic: many PhD students in a top institution underperform, and many PhD students in non-top institutions succeed. At the end of the day, the recognition of the institution plays very little role in your own success. However your area of interest is important: if there are no potential supervisors in your institution who are specialized in it, then it's reasonable to consider moving to a "more compatible" institution.
So my humble advice is:
(1) it is worth it to re-apply or should I just stick with where I am, and if so,
Not worth for the reputation of the institution; you should consider it only if you think you're not going to be able to work on your topic of interest with a good supervisor in your current institution.
(2) how do I ask for recommendation letters again without my professors feeling that I am wasting their time.
If you decide to do it, you can ask them if you can re-use their recent letters of recommendation.
(3)Is there a cost to re-applying again?
No idea but it should be easy to find out.
(4) What should I change this time?
If you decide to do it, you should probably explain that you now have a precise idea of the topic you plan to do and demonstrate why it makes sense for you to work on this topic at this institution, i.e. make it interesting for a supervisor to accept you.